Sec 2 G1 Maths Tutor Bukit Timah

Reader note: All shape sizes are original mathematical examples and not real building plans. Similarity and congruence rely on correct correspondence and assumptions, including positive scaling. Current tutoring venue, fees, subject-level placement and learning outcomes must be confirmed separately.

Bring one incorrect matching-side diagram to eduKate Singapore and ask how a tutor would teach corresponding sides and scale factors before expanding to harder geometry.

Official curriculum and reader notes

Bring one incorrect matching-side diagram to eduKate Singapore and ask how a tutor would teach corresponding sides and scale factors before expanding to harder geometry.

Official curriculum and reader notes

Bring one incorrect matching-side diagram to eduKate Singapore and ask how a tutor would teach corresponding sides and scale factors before expanding to harder geometry.

Day 1: identify congruent copies under translation, rotation and reflection.
Day 2: match corresponding sides in two named similar triangles.
Day 3: find new side lengths under whole-number and fractional scale factors.
Day 4: compare perimeter and area factors without memorising the wrong one.
Day 5: find a rectangle counterexample that shows identical angles alone are not sufficient for similarity.

Parents can ask ‘Which side matches which?’ and ‘Are you multiplying every length by the same factor?’ This tests the real idea more effectively than asking the child to repeat the word similar. If a pupil can explain why a square remains similar when resized but a stretched rectangle may not, the shape distinction is becoming independent.

Subject-level and examination-year accuracy

MOE’s 2020 Secondary Two G1 content covers congruent and similar figures and proportional corresponding sides. SEAB lists Mathematics K110 for the 2027 G1 SEC examination, referencing 4046 in earlier years. That 2027 specification belongs to its graduating candidates, not necessarily to a 2026 Secondary Two pupil’s next assessment. Follow the present school’s work and consult the official syllabus for the learner’s eventual graduating year.

Frequently asked questions

Are congruent shapes also similar?

Yes. Their similarity scale factor is one and all matching measurements agree.

Are all similar shapes congruent?

No. Similar shapes may differ in size; only those with scale factor one are congruent.

Does rotating a shape stop it being congruent?

No. Rotation preserves length and angle. So do translation and reflection.

Why is area factor k²?

Area has two dimensions, both scaling by k. The combined factor is k×k=k².

Are all rectangles similar?

No. Their corresponding side ratios must agree; rectangles can have different aspect ratios.

What should a tutor check beyond the final side answer?

The matched vertex order, scale direction, uniform side ratios and whether the conclusion is similarity or congruence.

Official curriculum and reader notes

Bring one incorrect matching-side diagram to eduKate Singapore and ask how a tutor would teach corresponding sides and scale factors before expanding to harder geometry.

Day 1: identify congruent copies under translation, rotation and reflection.
Day 2: match corresponding sides in two named similar triangles.
Day 3: find new side lengths under whole-number and fractional scale factors.
Day 4: compare perimeter and area factors without memorising the wrong one.
Day 5: find a rectangle counterexample that shows identical angles alone are not sufficient for similarity.

Subject-level and examination-year accuracy

Frequently asked questions

Are congruent shapes also similar?

Yes. Their similarity scale factor is one and all matching measurements agree.

Are all similar shapes congruent?

No. Similar shapes may differ in size; only those with scale factor one are congruent.

Does rotating a shape stop it being congruent?

No. Rotation preserves length and angle. So do translation and reflection.

Why is area factor k²?

Area has two dimensions, both scaling by k. The combined factor is k×k=k².

Are all rectangles similar?

No. Their corresponding side ratios must agree; rectangles can have different aspect ratios.

What should a tutor check beyond the final side answer?

The matched vertex order, scale direction, uniform side ratios and whether the conclusion is similarity or congruence.

Official curriculum and reader notes

Bring one incorrect matching-side diagram to eduKate Singapore and ask how a tutor would teach corresponding sides and scale factors before expanding to harder geometry.

10 minutes: retrieve equal angles, parallel marks and common ratios.
20 minutes: compare a congruent and a scaled triangle with labelled correspondence.
25 minutes: calculate missing sides and check matching ratios independently.
15 minutes: compare perimeter versus area scaling and rectangular aspect ratios.
10 minutes: solve an unseen side correspondence or counterexample question.
10 minutes: explain the first mismatch and set a later retest.

Five short practice sessions

Day 1: identify congruent copies under translation, rotation and reflection.
Day 2: match corresponding sides in two named similar triangles.
Day 3: find new side lengths under whole-number and fractional scale factors.
Day 4: compare perimeter and area factors without memorising the wrong one.
Day 5: find a rectangle counterexample that shows identical angles alone are not sufficient for similarity.

Subject-level and examination-year accuracy

Frequently asked questions

Are congruent shapes also similar?

Yes. Their similarity scale factor is one and all matching measurements agree.

Are all similar shapes congruent?

No. Similar shapes may differ in size; only those with scale factor one are congruent.

Does rotating a shape stop it being congruent?

No. Rotation preserves length and angle. So do translation and reflection.

Why is area factor k²?

Area has two dimensions, both scaling by k. The combined factor is k×k=k².

Are all rectangles similar?

No. Their corresponding side ratios must agree; rectangles can have different aspect ratios.

What should a tutor check beyond the final side answer?

The matched vertex order, scale direction, uniform side ratios and whether the conclusion is similarity or congruence.

Official curriculum and reader notes

Bring one incorrect matching-side diagram to eduKate Singapore and ask how a tutor would teach corresponding sides and scale factors before expanding to harder geometry.

10 minutes: retrieve equal angles, parallel marks and common ratios.
20 minutes: compare a congruent and a scaled triangle with labelled correspondence.
25 minutes: calculate missing sides and check matching ratios independently.
15 minutes: compare perimeter versus area scaling and rectangular aspect ratios.
10 minutes: solve an unseen side correspondence or counterexample question.
10 minutes: explain the first mismatch and set a later retest.

Five short practice sessions

Day 1: identify congruent copies under translation, rotation and reflection.
Day 2: match corresponding sides in two named similar triangles.
Day 3: find new side lengths under whole-number and fractional scale factors.
Day 4: compare perimeter and area factors without memorising the wrong one.
Day 5: find a rectangle counterexample that shows identical angles alone are not sufficient for similarity.

Subject-level and examination-year accuracy

Frequently asked questions

Are congruent shapes also similar?

Yes. Their similarity scale factor is one and all matching measurements agree.

Are all similar shapes congruent?

No. Similar shapes may differ in size; only those with scale factor one are congruent.

Does rotating a shape stop it being congruent?

No. Rotation preserves length and angle. So do translation and reflection.

Why is area factor k²?

Area has two dimensions, both scaling by k. The combined factor is k×k=k².

Are all rectangles similar?

No. Their corresponding side ratios must agree; rectangles can have different aspect ratios.

What should a tutor check beyond the final side answer?

The matched vertex order, scale direction, uniform side ratios and whether the conclusion is similarity or congruence.

Official curriculum and reader notes

Bring one incorrect matching-side diagram to eduKate Singapore and ask how a tutor would teach corresponding sides and scale factors before expanding to harder geometry.

One-minute diagnostic
What congruent and similar mean
Find corresponding sides
Length, perimeter and area factors
Original worked shape clinics
Four common mistakes
Three-pupil tutorial model
Home practice
Frequently asked questions
Official sources

The one-minute comparison

Ask a student whether triangles with sides 3,4,5 and 6,8,10 are similar, and whether they are congruent. They are similar at scale factor two but not congruent. Then ask what happens to area: it becomes four times as large. Finally ask whether two 4-by-6 and 5-by-7 rectangles must be similar. No: the side ratios differ. These three questions separate the words, the scaling rule and the false assumption that all rectangles have the same shape.

Similarity preserves shape, congruence preserves size too

Congruence tolerates movement and reflection

Two congruent figures need not occupy the same coordinates on a page. One may be rotated, translated or reflected to overlap the other. The matching lengths and angles stay equal. If a pupil rules out congruence because a triangle points down instead of up, use tracing paper or a mental rotation to distinguish orientation from shape and size.

Similarity requires one multiplier for matching lengths

For triangles with sides 3,4,5 and 6,8,10, divide the new matched side by the old for each pair: 6/3=2, 8/4=2, 10/5=2. The common factor two supports similarity. A pair of arbitrary rectangles with side lengths 3 by 4 and 6 by 9 does not share one factor, because width doubled but height multiplied by 2.25. The words ‘both are rectangles’ do not repair the mismatch.

Match corresponding vertices before dividing

If triangle ABC is stated similar to triangle DEF in that order, A matches D, B matches E and C matches F. Thus AB matches DE, BC matches EF and AC matches DF. Ratios of arbitrary sides that do not correspond can look inconsistent even for valid similar shapes. Mark the three matched side pairs with matching colours and keep the same direction—new divided by old—throughout the calculation.

Why an area does not scale like a side length

If a rectangle’s length and width both double, area becomes four times its earlier value. A 3-by-4 rectangle has area 12; a 6-by-8 similar rectangle has area 48. The perimeter, however, doubles from 14 to 28. In general if corresponding lengths multiply by k, perimeter multiplies by k and area by k². Work backwards from an area ratio by taking the positive square root of the area scale factor.

This difference has real applications in drawings, maps and display resizing. But a mathematical scale drawing is an ideal model: if one width is changed while height stays fixed, the figure has been stretched and its aspect ratio changed. A pupil should not automatically call every enlarged image similar. The same uniform multiplier must govern both dimensions.

125 original correspondence and scale-factor clinics

These questions move from triangles with corresponding sides to missing-side calculations, rectangle aspect ratios and explanation-based counterexamples. All are original teaching examples, not reproduced examination questions. A few precise comparisons are more useful than attempting every clinic in one sitting. First name the matching side pairs, then calculate the scale factor, then decide similarity or congruence.

1. Similar-triangle comparison 1: factor 2/1

Try this. Triangle A has side lengths 3, 4, 5 cm and triangle B has the corresponding side lengths 6, 8, 10 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 6/3, 8/4, 10/5, each equal to 2. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 12 cm; the new perimeter is 24 cm, exactly 2 times as large.

First possible misconception. An equal-angle diagram does not automatically mean equal side lengths. ‘Similar’ means the shape agrees up to a common scale, while ‘congruent’ needs the matching lengths to be the same. A student who compares the wrong side labels may falsely reject the similarity even when corresponding ratios match.

Independent check. The perimeter ratio is 24/12=2. All three matched lengths change by that same factor. The area ratio would be 4, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 4, its perimeter would be 48 cm, but area compared with A would scale by 16. Ask why perimeter and area factors differ without using a memorised rule alone.

2. Similar-triangle comparison 2: factor 3/1

Try this. Triangle A has side lengths 5, 12, 13 cm and triangle B has the corresponding side lengths 15, 36, 39 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 15/5, 36/12, 39/13, each equal to 3. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 30 cm; the new perimeter is 90 cm, exactly 3 times as large.

Independent check. The perimeter ratio is 90/30=3. All three matched lengths change by that same factor. The area ratio would be 9, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 6, its perimeter would be 180 cm, but area compared with A would scale by 36. Ask why perimeter and area factors differ without using a memorised rule alone.

3. Similar-triangle comparison 3: factor 4/1

Try this. Triangle A has side lengths 6, 8, 10 cm and triangle B has the corresponding side lengths 24, 32, 40 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 24/6, 32/8, 40/10, each equal to 4. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 24 cm; the new perimeter is 96 cm, exactly 4 times as large.

Independent check. The perimeter ratio is 96/24=4. All three matched lengths change by that same factor. The area ratio would be 16, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 8, its perimeter would be 192 cm, but area compared with A would scale by 64. Ask why perimeter and area factors differ without using a memorised rule alone.

4. Similar-triangle comparison 4: factor 1/2

Try this. Triangle A has side lengths 7, 24, 25 cm and triangle B has the corresponding side lengths 3.5, 12, 12.5 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 3.5/7, 12/24, 12.5/25, each equal to 0.5. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 56 cm; the new perimeter is 28 cm, exactly 0.5 times as large.

Independent check. The perimeter ratio is 28/56=0.5. All three matched lengths change by that same factor. The area ratio would be 0.25, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 1, its perimeter would be 56 cm, but area compared with A would scale by 1. Ask why perimeter and area factors differ without using a memorised rule alone.

5. Similar-triangle comparison 5: factor 3/2

Try this. Triangle A has side lengths 8, 15, 17 cm and triangle B has the corresponding side lengths 12, 22.5, 25.5 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 12/8, 22.5/15, 25.5/17, each equal to 1.5. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 40 cm; the new perimeter is 60 cm, exactly 1.5 times as large.

Independent check. The perimeter ratio is 60/40=1.5. All three matched lengths change by that same factor. The area ratio would be 2.25, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 3, its perimeter would be 120 cm, but area compared with A would scale by 9. Ask why perimeter and area factors differ without using a memorised rule alone.

6. Similar-triangle comparison 6: factor 5/2

Try this. Triangle A has side lengths 9, 12, 15 cm and triangle B has the corresponding side lengths 22.5, 30, 37.5 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 22.5/9, 30/12, 37.5/15, each equal to 2.5. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 36 cm; the new perimeter is 90 cm, exactly 2.5 times as large.

Independent check. The perimeter ratio is 90/36=2.5. All three matched lengths change by that same factor. The area ratio would be 6.25, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 5, its perimeter would be 180 cm, but area compared with A would scale by 25. Ask why perimeter and area factors differ without using a memorised rule alone.

7. Similar-triangle comparison 7: factor 1/3

Try this. Triangle A has sides 10, 24, 26 cm and triangle B has corresponding sides 10/3, 8, 26/3 cm (exact fractions are shown rather than prematurely rounded decimals). Decide whether they are similar or congruent, and compare perimeters.

Worked reasoning. The new-to-old ratios are (10/3)/10, (8)/24 and (26/3)/26. Each simplifies exactly to 1/3. Thus the triangles are similar. Because 1/3 is not one, they are not congruent. The original perimeter is 60 cm and the new perimeter is 20 cm, scaled by the same length factor 1/3.

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 1/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

Independent check. The perimeter ratio (20)/60 simplifies to 1/3, while the area ratio is 1/9. A length multiplier and an area multiplier answer different questions; the area factor is the square of 1/3.

Transfer. A third triangle enlarged from A by length factor 2/3 would have perimeter 40 cm and area factor 4/9 compared with A. Rebuild these results from proportional side lengths before attempting a decimal approximation.

8. Similar-triangle comparison 8: factor 4/3

Try this. Triangle A has sides 12, 16, 20 cm and triangle B has corresponding sides 16, 64/3, 80/3 cm (exact fractions are shown rather than prematurely rounded decimals). Decide whether they are similar or congruent, and compare perimeters.

Worked reasoning. The new-to-old ratios are (16)/12, (64/3)/16 and (80/3)/20. Each simplifies exactly to 4/3. Thus the triangles are similar. Because 4/3 is not one, they are not congruent. The original perimeter is 48 cm and the new perimeter is 64 cm, scaled by the same length factor 4/3.

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 4/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

Independent check. The perimeter ratio (64)/48 simplifies to 4/3, while the area ratio is 16/9. A length multiplier and an area multiplier answer different questions; the area factor is the square of 4/3.

Transfer. A third triangle enlarged from A by length factor 8/3 would have perimeter 128 cm and area factor 64/9 compared with A. Rebuild these results from proportional side lengths before attempting a decimal approximation.

9. Similar-triangle comparison 9: factor 6/5

Try this. Triangle A has side lengths 9, 40, 41 cm and triangle B has the corresponding side lengths 10.8, 48, 49.2 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 10.8/9, 48/40, 49.2/41, each equal to 1.2. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 90 cm; the new perimeter is 108 cm, exactly 1.2 times as large.

Independent check. The perimeter ratio is 108/90=1.2. All three matched lengths change by that same factor. The area ratio would be 1.44, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 2.4, its perimeter would be 216 cm, but area compared with A would scale by 5.76. Ask why perimeter and area factors differ without using a memorised rule alone.

10. Similar-triangle comparison 10: factor 7/4

Try this. Triangle A has side lengths 15, 20, 25 cm and triangle B has the corresponding side lengths 26.25, 35, 43.75 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 26.25/15, 35/20, 43.75/25, each equal to 1.75. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 60 cm; the new perimeter is 105 cm, exactly 1.75 times as large.

Independent check. The perimeter ratio is 105/60=1.75. All three matched lengths change by that same factor. The area ratio would be 3.0625, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 3.5, its perimeter would be 210 cm, but area compared with A would scale by 12.25. Ask why perimeter and area factors differ without using a memorised rule alone.

11. Similar-triangle comparison 11: factor 5/4

Try this. Triangle A has side lengths 5, 5, 6 cm and triangle B has the corresponding side lengths 6.25, 6.25, 7.5 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 6.25/5, 6.25/5, 7.5/6, each equal to 1.25. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 16 cm; the new perimeter is 20 cm, exactly 1.25 times as large.

Independent check. The perimeter ratio is 20/16=1.25. All three matched lengths change by that same factor. The area ratio would be 1.5625, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 2.5, its perimeter would be 40 cm, but area compared with A would scale by 6.25. Ask why perimeter and area factors differ without using a memorised rule alone.

12. Similar-triangle comparison 12: factor 1/1

Try this. Triangle A has side lengths 4, 5, 6 cm and triangle B has the corresponding side lengths 4, 5, 6 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 4/4, 5/5, 6/6, each equal to 1. This common side ratio establishes similarity for these valid triangles. Because the factor is one, the figures are congruent as well. The old perimeter is 15 cm; the new perimeter is 15 cm, exactly 1 times as large.

Independent check. The perimeter ratio is 15/15=1. All three matched lengths change by that same factor. The area ratio would be 1, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 2, its perimeter would be 30 cm, but area compared with A would scale by 4. Ask why perimeter and area factors differ without using a memorised rule alone.

13. Similar-triangle comparison 13: factor 2/3

Try this. Triangle A has sides 3, 5, 7 cm and triangle B has corresponding sides 2, 10/3, 14/3 cm (exact fractions are shown rather than prematurely rounded decimals). Decide whether they are similar or congruent, and compare perimeters.

Worked reasoning. The new-to-old ratios are (2)/3, (10/3)/5 and (14/3)/7. Each simplifies exactly to 2/3. Thus the triangles are similar. Because 2/3 is not one, they are not congruent. The original perimeter is 15 cm and the new perimeter is 10 cm, scaled by the same length factor 2/3.

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 2/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

Independent check. The perimeter ratio (10)/15 simplifies to 2/3, while the area ratio is 4/9. A length multiplier and an area multiplier answer different questions; the area factor is the square of 2/3.

Transfer. A third triangle enlarged from A by length factor 4/3 would have perimeter 20 cm and area factor 16/9 compared with A. Rebuild these results from proportional side lengths before attempting a decimal approximation.

14. Similar-triangle comparison 14: factor 2/1

Try this. Triangle A has side lengths 6, 7, 8 cm and triangle B has the corresponding side lengths 12, 14, 16 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 12/6, 14/7, 16/8, each equal to 2. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 21 cm; the new perimeter is 42 cm, exactly 2 times as large.

Independent check. The perimeter ratio is 42/21=2. All three matched lengths change by that same factor. The area ratio would be 4, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 4, its perimeter would be 84 cm, but area compared with A would scale by 16. Ask why perimeter and area factors differ without using a memorised rule alone.

15. Similar-triangle comparison 15: factor 3/1

Try this. Triangle A has side lengths 4, 4, 7 cm and triangle B has the corresponding side lengths 12, 12, 21 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 12/4, 12/4, 21/7, each equal to 3. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 15 cm; the new perimeter is 45 cm, exactly 3 times as large.

Independent check. The perimeter ratio is 45/15=3. All three matched lengths change by that same factor. The area ratio would be 9, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 6, its perimeter would be 90 cm, but area compared with A would scale by 36. Ask why perimeter and area factors differ without using a memorised rule alone.

16. Similar-triangle comparison 16: factor 4/1

Try this. Triangle A has side lengths 3, 4, 5 cm and triangle B has the corresponding side lengths 12, 16, 20 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 12/3, 16/4, 20/5, each equal to 4. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 12 cm; the new perimeter is 48 cm, exactly 4 times as large.

Independent check. The perimeter ratio is 48/12=4. All three matched lengths change by that same factor. The area ratio would be 16, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 8, its perimeter would be 96 cm, but area compared with A would scale by 64. Ask why perimeter and area factors differ without using a memorised rule alone.

17. Similar-triangle comparison 17: factor 1/2

Try this. Triangle A has side lengths 5, 12, 13 cm and triangle B has the corresponding side lengths 2.5, 6, 6.5 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 2.5/5, 6/12, 6.5/13, each equal to 0.5. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 30 cm; the new perimeter is 15 cm, exactly 0.5 times as large.

Independent check. The perimeter ratio is 15/30=0.5. All three matched lengths change by that same factor. The area ratio would be 0.25, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 1, its perimeter would be 30 cm, but area compared with A would scale by 1. Ask why perimeter and area factors differ without using a memorised rule alone.

18. Similar-triangle comparison 18: factor 3/2

Try this. Triangle A has side lengths 6, 8, 10 cm and triangle B has the corresponding side lengths 9, 12, 15 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 9/6, 12/8, 15/10, each equal to 1.5. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 24 cm; the new perimeter is 36 cm, exactly 1.5 times as large.

Independent check. The perimeter ratio is 36/24=1.5. All three matched lengths change by that same factor. The area ratio would be 2.25, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 3, its perimeter would be 72 cm, but area compared with A would scale by 9. Ask why perimeter and area factors differ without using a memorised rule alone.

19. Similar-triangle comparison 19: factor 5/2

Try this. Triangle A has side lengths 7, 24, 25 cm and triangle B has the corresponding side lengths 17.5, 60, 62.5 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 17.5/7, 60/24, 62.5/25, each equal to 2.5. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 56 cm; the new perimeter is 140 cm, exactly 2.5 times as large.

Independent check. The perimeter ratio is 140/56=2.5. All three matched lengths change by that same factor. The area ratio would be 6.25, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 5, its perimeter would be 280 cm, but area compared with A would scale by 25. Ask why perimeter and area factors differ without using a memorised rule alone.

20. Similar-triangle comparison 20: factor 1/3

Try this. Triangle A has sides 8, 15, 17 cm and triangle B has corresponding sides 8/3, 5, 17/3 cm (exact fractions are shown rather than prematurely rounded decimals). Decide whether they are similar or congruent, and compare perimeters.

Worked reasoning. The new-to-old ratios are (8/3)/8, (5)/15 and (17/3)/17. Each simplifies exactly to 1/3. Thus the triangles are similar. Because 1/3 is not one, they are not congruent. The original perimeter is 40 cm and the new perimeter is 40/3 cm, scaled by the same length factor 1/3.

Independent check. The perimeter ratio (40/3)/40 simplifies to 1/3, while the area ratio is 1/9. A length multiplier and an area multiplier answer different questions; the area factor is the square of 1/3.

Transfer. A third triangle enlarged from A by length factor 2/3 would have perimeter 80/3 cm and area factor 4/9 compared with A. Rebuild these results from proportional side lengths before attempting a decimal approximation.

21. Similar-triangle comparison 21: factor 4/3

Try this. Triangle A has side lengths 9, 12, 15 cm and triangle B has the corresponding side lengths 12, 16, 20 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 12/9, 16/12, 20/15, each equal to 1.3333. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 36 cm; the new perimeter is 48 cm, exactly 1.3333 times as large.

Independent check. The perimeter ratio is 48/36=1.3333. All three matched lengths change by that same factor. The area ratio would be 1.7778, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 2.6667, its perimeter would be 96 cm, but area compared with A would scale by 7.1111. Ask why perimeter and area factors differ without using a memorised rule alone.

22. Similar-triangle comparison 22: factor 6/5

Try this. Triangle A has side lengths 10, 24, 26 cm and triangle B has the corresponding side lengths 12, 28.8, 31.2 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 12/10, 28.8/24, 31.2/26, each equal to 1.2. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 60 cm; the new perimeter is 72 cm, exactly 1.2 times as large.

Independent check. The perimeter ratio is 72/60=1.2. All three matched lengths change by that same factor. The area ratio would be 1.44, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 2.4, its perimeter would be 144 cm, but area compared with A would scale by 5.76. Ask why perimeter and area factors differ without using a memorised rule alone.

23. Similar-triangle comparison 23: factor 7/4

Try this. Triangle A has side lengths 12, 16, 20 cm and triangle B has the corresponding side lengths 21, 28, 35 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 21/12, 28/16, 35/20, each equal to 1.75. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 48 cm; the new perimeter is 84 cm, exactly 1.75 times as large.

Independent check. The perimeter ratio is 84/48=1.75. All three matched lengths change by that same factor. The area ratio would be 3.0625, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 3.5, its perimeter would be 168 cm, but area compared with A would scale by 12.25. Ask why perimeter and area factors differ without using a memorised rule alone.

24. Similar-triangle comparison 24: factor 5/4

Try this. Triangle A has side lengths 9, 40, 41 cm and triangle B has the corresponding side lengths 11.25, 50, 51.25 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 11.25/9, 50/40, 51.25/41, each equal to 1.25. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 90 cm; the new perimeter is 112.5 cm, exactly 1.25 times as large.

Independent check. The perimeter ratio is 112.5/90=1.25. All three matched lengths change by that same factor. The area ratio would be 1.5625, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 2.5, its perimeter would be 225 cm, but area compared with A would scale by 6.25. Ask why perimeter and area factors differ without using a memorised rule alone.

25. Similar-triangle comparison 25: factor 1/1

Try this. Triangle A has side lengths 15, 20, 25 cm and triangle B has the corresponding side lengths 15, 20, 25 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 15/15, 20/20, 25/25, each equal to 1. This common side ratio establishes similarity for these valid triangles. Because the factor is one, the figures are congruent as well. The old perimeter is 60 cm; the new perimeter is 60 cm, exactly 1 times as large.

Independent check. The perimeter ratio is 60/60=1. All three matched lengths change by that same factor. The area ratio would be 1, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 2, its perimeter would be 120 cm, but area compared with A would scale by 4. Ask why perimeter and area factors differ without using a memorised rule alone.

26. Similar-triangle comparison 26: factor 2/3

Try this. Triangle A has sides 5, 5, 6 cm and triangle B has corresponding sides 10/3, 10/3, 4 cm (exact fractions are shown rather than prematurely rounded decimals). Decide whether they are similar or congruent, and compare perimeters.

Worked reasoning. The new-to-old ratios are (10/3)/5, (10/3)/5 and (4)/6. Each simplifies exactly to 2/3. Thus the triangles are similar. Because 2/3 is not one, they are not congruent. The original perimeter is 16 cm and the new perimeter is 32/3 cm, scaled by the same length factor 2/3.

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 2/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

Independent check. The perimeter ratio (32/3)/16 simplifies to 2/3, while the area ratio is 4/9. A length multiplier and an area multiplier answer different questions; the area factor is the square of 2/3.

Transfer. A third triangle enlarged from A by length factor 4/3 would have perimeter 64/3 cm and area factor 16/9 compared with A. Rebuild these results from proportional side lengths before attempting a decimal approximation.

27. Similar-triangle comparison 27: factor 2/1

Try this. Triangle A has side lengths 4, 5, 6 cm and triangle B has the corresponding side lengths 8, 10, 12 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 8/4, 10/5, 12/6, each equal to 2. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 15 cm; the new perimeter is 30 cm, exactly 2 times as large.

Independent check. The perimeter ratio is 30/15=2. All three matched lengths change by that same factor. The area ratio would be 4, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 4, its perimeter would be 60 cm, but area compared with A would scale by 16. Ask why perimeter and area factors differ without using a memorised rule alone.

28. Similar-triangle comparison 28: factor 3/1

Try this. Triangle A has side lengths 3, 5, 7 cm and triangle B has the corresponding side lengths 9, 15, 21 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 9/3, 15/5, 21/7, each equal to 3. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 15 cm; the new perimeter is 45 cm, exactly 3 times as large.

29. Similar-triangle comparison 29: factor 4/1

Try this. Triangle A has side lengths 6, 7, 8 cm and triangle B has the corresponding side lengths 24, 28, 32 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 24/6, 28/7, 32/8, each equal to 4. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 21 cm; the new perimeter is 84 cm, exactly 4 times as large.

Independent check. The perimeter ratio is 84/21=4. All three matched lengths change by that same factor. The area ratio would be 16, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 8, its perimeter would be 168 cm, but area compared with A would scale by 64. Ask why perimeter and area factors differ without using a memorised rule alone.

30. Similar-triangle comparison 30: factor 1/2

Try this. Triangle A has side lengths 4, 4, 7 cm and triangle B has the corresponding side lengths 2, 2, 3.5 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 2/4, 2/4, 3.5/7, each equal to 0.5. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 15 cm; the new perimeter is 7.5 cm, exactly 0.5 times as large.

Independent check. The perimeter ratio is 7.5/15=0.5. All three matched lengths change by that same factor. The area ratio would be 0.25, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 1, its perimeter would be 15 cm, but area compared with A would scale by 1. Ask why perimeter and area factors differ without using a memorised rule alone.

31. Similar-triangle comparison 31: factor 3/2

Try this. Triangle A has side lengths 3, 4, 5 cm and triangle B has the corresponding side lengths 4.5, 6, 7.5 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 4.5/3, 6/4, 7.5/5, each equal to 1.5. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 12 cm; the new perimeter is 18 cm, exactly 1.5 times as large.

Independent check. The perimeter ratio is 18/12=1.5. All three matched lengths change by that same factor. The area ratio would be 2.25, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 3, its perimeter would be 36 cm, but area compared with A would scale by 9. Ask why perimeter and area factors differ without using a memorised rule alone.

32. Similar-triangle comparison 32: factor 5/2

Try this. Triangle A has side lengths 5, 12, 13 cm and triangle B has the corresponding side lengths 12.5, 30, 32.5 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 12.5/5, 30/12, 32.5/13, each equal to 2.5. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 30 cm; the new perimeter is 75 cm, exactly 2.5 times as large.

Independent check. The perimeter ratio is 75/30=2.5. All three matched lengths change by that same factor. The area ratio would be 6.25, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 5, its perimeter would be 150 cm, but area compared with A would scale by 25. Ask why perimeter and area factors differ without using a memorised rule alone.

33. Similar-triangle comparison 33: factor 1/3

Try this. Triangle A has sides 6, 8, 10 cm and triangle B has corresponding sides 2, 8/3, 10/3 cm (exact fractions are shown rather than prematurely rounded decimals). Decide whether they are similar or congruent, and compare perimeters.

Worked reasoning. The new-to-old ratios are (2)/6, (8/3)/8 and (10/3)/10. Each simplifies exactly to 1/3. Thus the triangles are similar. Because 1/3 is not one, they are not congruent. The original perimeter is 24 cm and the new perimeter is 8 cm, scaled by the same length factor 1/3.

Independent check. The perimeter ratio (8)/24 simplifies to 1/3, while the area ratio is 1/9. A length multiplier and an area multiplier answer different questions; the area factor is the square of 1/3.

Transfer. A third triangle enlarged from A by length factor 2/3 would have perimeter 16 cm and area factor 4/9 compared with A. Rebuild these results from proportional side lengths before attempting a decimal approximation.

34. Similar-triangle comparison 34: factor 4/3

Try this. Triangle A has sides 7, 24, 25 cm and triangle B has corresponding sides 28/3, 32, 100/3 cm (exact fractions are shown rather than prematurely rounded decimals). Decide whether they are similar or congruent, and compare perimeters.

Worked reasoning. The new-to-old ratios are (28/3)/7, (32)/24 and (100/3)/25. Each simplifies exactly to 4/3. Thus the triangles are similar. Because 4/3 is not one, they are not congruent. The original perimeter is 56 cm and the new perimeter is 224/3 cm, scaled by the same length factor 4/3.

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 4/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

Independent check. The perimeter ratio (224/3)/56 simplifies to 4/3, while the area ratio is 16/9. A length multiplier and an area multiplier answer different questions; the area factor is the square of 4/3.

Transfer. A third triangle enlarged from A by length factor 8/3 would have perimeter 448/3 cm and area factor 64/9 compared with A. Rebuild these results from proportional side lengths before attempting a decimal approximation.

35. Similar-triangle comparison 35: factor 6/5

Try this. Triangle A has side lengths 8, 15, 17 cm and triangle B has the corresponding side lengths 9.6, 18, 20.4 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 9.6/8, 18/15, 20.4/17, each equal to 1.2. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 40 cm; the new perimeter is 48 cm, exactly 1.2 times as large.

Independent check. The perimeter ratio is 48/40=1.2. All three matched lengths change by that same factor. The area ratio would be 1.44, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 2.4, its perimeter would be 96 cm, but area compared with A would scale by 5.76. Ask why perimeter and area factors differ without using a memorised rule alone.

36. Similar-triangle comparison 36: factor 7/4

Try this. Triangle A has side lengths 9, 12, 15 cm and triangle B has the corresponding side lengths 15.75, 21, 26.25 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 15.75/9, 21/12, 26.25/15, each equal to 1.75. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 36 cm; the new perimeter is 63 cm, exactly 1.75 times as large.

Independent check. The perimeter ratio is 63/36=1.75. All three matched lengths change by that same factor. The area ratio would be 3.0625, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 3.5, its perimeter would be 126 cm, but area compared with A would scale by 12.25. Ask why perimeter and area factors differ without using a memorised rule alone.

37. Similar-triangle comparison 37: factor 5/4

Try this. Triangle A has side lengths 10, 24, 26 cm and triangle B has the corresponding side lengths 12.5, 30, 32.5 cm in exactly the same listed order. Determine whether the triangles are similar, whether they are congruent, and how their perimeters compare.

Worked reasoning. The corresponding side quotients from B to A are 12.5/10, 30/24, 32.5/26, each equal to 1.25. This common side ratio establishes similarity for these valid triangles. Because the factor is not one, the triangles differ in size and are not congruent. The old perimeter is 60 cm; the new perimeter is 75 cm, exactly 1.25 times as large.

Independent check. The perimeter ratio is 75/60=1.25. All three matched lengths change by that same factor. The area ratio would be 1.5625, because two linear dimensions scale, even if the triangle’s height was not stated.

Transfer. If a third copy enlarged triangle A by factor 2.5, its perimeter would be 150 cm, but area compared with A would scale by 6.25. Ask why perimeter and area factors differ without using a memorised rule alone.

38. Missing corresponding side 1: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 3, 4, 5 cm. Triangle B’s side matching the 3-cm side is 12 cm. Find the B-side matching A’s 4-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=12/3=4. Multiply the next corresponding A-side 4 cm by this factor to obtain 16 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

First possible misconception. Dividing the original side by the new one would compute the reverse scale factor. The direction ‘from A to B’ matters. Also, without the given similarity premise, a single matching side pair is not enough to prove two arbitrary triangles are similar.

Independent check. Reverse the conversion: 16/4=4 cm. The B perimeter would be 48 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.25, not 4. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

39. Missing corresponding side 2: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 7, 24, 25 cm. Triangle B’s side matching the 24-cm side is 12 cm. Find the B-side matching A’s 25-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=12/24=0.5. Multiply the next corresponding A-side 25 cm by this factor to obtain 12.5 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 12.5/0.5=25 cm. The B perimeter would be 28 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 2, not 0.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

40. Missing corresponding side 3: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 10, 24, 26 cm. Triangle B’s side matching the 26-cm side is 39 cm. Find the B-side matching A’s 10-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=39/26=1.5. Multiply the next corresponding A-side 10 cm by this factor to obtain 15 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 15/1.5=10 cm. The B perimeter would be 90 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.6667, not 1.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

41. Missing corresponding side 4: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 15, 20, 25 cm. Triangle B’s side matching the 15-cm side is 37.5 cm. Find the B-side matching A’s 20-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=37.5/15=2.5. Multiply the next corresponding A-side 20 cm by this factor to obtain 50 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 50/2.5=20 cm. The B perimeter would be 150 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.4, not 2.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

42. Missing corresponding side 5: exact scale factor

Try this. Triangle A has sides 3, 5, 7 cm. A similar triangle B has a side of 5/3 cm corresponding to A’s 5-cm side. Find B’s side corresponding to A’s 7-cm side. The lengths are given exactly as fractions where necessary.

Worked reasoning. The new-to-old scale factor is (5/3)/5 = 1/3. Multiply A’s 7 cm by 1/3: the required B side is 7/3 cm. Similarity is a stated fact in the question, so the single known matched pair determines its positive scale factor.

First possible misconception. The reverse fraction 3 would scale B back to A. Do not reverse the order of the lengths unintentionally. An exact one-third scale can produce lengths such as 5/3 cm, which should not be silently displayed as 1.6667 cm and then called exactly proportional.

Independent check. Reverse the result: (7/3)÷(1/3)=7 cm. The new perimeter would be 5 cm compared with original 15 cm, again preserving scale factor 1/3.

Transfer. Reverse the enlargement or reduction from B to A using factor 3. The factor in the opposite direction is the reciprocal. Ask the learner to name the two matched sides before calculating.

43. Missing corresponding side 6: exact scale factor

Try this. Triangle A has sides 3, 4, 5 cm. A similar triangle B has a side of 20/3 cm corresponding to A’s 5-cm side. Find B’s side corresponding to A’s 3-cm side. The lengths are given exactly as fractions where necessary.

Worked reasoning. The new-to-old scale factor is (20/3)/5 = 4/3. Multiply A’s 3 cm by 4/3: the required B side is 4 cm. Similarity is a stated fact in the question, so the single known matched pair determines its positive scale factor.

First possible misconception. The reverse fraction 3/4 would scale B back to A. Do not reverse the order of the lengths unintentionally. An exact one-third scale can produce lengths such as 5/3 cm, which should not be silently displayed as 1.6667 cm and then called exactly proportional.

Independent check. Reverse the result: (4)÷(4/3)=3 cm. The new perimeter would be 16 cm compared with original 12 cm, again preserving scale factor 4/3.

Transfer. Reverse the enlargement or reduction from B to A using factor 3/4. The factor in the opposite direction is the reciprocal. Ask the learner to name the two matched sides before calculating.

44. Missing corresponding side 7: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 7, 24, 25 cm. Triangle B’s side matching the 7-cm side is 8.4 cm. Find the B-side matching A’s 24-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=8.4/7=1.2. Multiply the next corresponding A-side 24 cm by this factor to obtain 28.8 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 28.8/1.2=24 cm. The B perimeter would be 67.2 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8333, not 1.2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

45. Missing corresponding side 8: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 10, 24, 26 cm. Triangle B’s side matching the 24-cm side is 42 cm. Find the B-side matching A’s 26-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=42/24=1.75. Multiply the next corresponding A-side 26 cm by this factor to obtain 45.5 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 45.5/1.75=26 cm. The B perimeter would be 105 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5714, not 1.75. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

46. Missing corresponding side 9: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 15, 20, 25 cm. Triangle B’s side matching the 25-cm side is 31.25 cm. Find the B-side matching A’s 15-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=31.25/25=1.25. Multiply the next corresponding A-side 15 cm by this factor to obtain 18.75 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 18.75/1.25=15 cm. The B perimeter would be 75 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8, not 1.25. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

47. Missing corresponding side 10: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 3, 5, 7 cm. Triangle B’s side matching the 3-cm side is 3 cm. Find the B-side matching A’s 5-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=3/3=1. Multiply the next corresponding A-side 5 cm by this factor to obtain 5 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 5/1=5 cm. The B perimeter would be 15 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 1, not 1. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

48. Missing corresponding side 11: exact scale factor

Try this. Triangle A has sides 3, 4, 5 cm. A similar triangle B has a side of 8/3 cm corresponding to A’s 4-cm side. Find B’s side corresponding to A’s 5-cm side. The lengths are given exactly as fractions where necessary.

Worked reasoning. The new-to-old scale factor is (8/3)/4 = 2/3. Multiply A’s 5 cm by 2/3: the required B side is 10/3 cm. Similarity is a stated fact in the question, so the single known matched pair determines its positive scale factor.

First possible misconception. The reverse fraction 3/2 would scale B back to A. Do not reverse the order of the lengths unintentionally. An exact one-third scale can produce lengths such as 5/3 cm, which should not be silently displayed as 1.6667 cm and then called exactly proportional.

Independent check. Reverse the result: (10/3)÷(2/3)=5 cm. The new perimeter would be 8 cm compared with original 12 cm, again preserving scale factor 2/3.

Transfer. Reverse the enlargement or reduction from B to A using factor 3/2. The factor in the opposite direction is the reciprocal. Ask the learner to name the two matched sides before calculating.

49. Missing corresponding side 12: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 7, 24, 25 cm. Triangle B’s side matching the 25-cm side is 50 cm. Find the B-side matching A’s 7-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=50/25=2. Multiply the next corresponding A-side 7 cm by this factor to obtain 14 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 14/2=7 cm. The B perimeter would be 112 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5, not 2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

50. Missing corresponding side 13: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 10, 24, 26 cm. Triangle B’s side matching the 10-cm side is 30 cm. Find the B-side matching A’s 24-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=30/10=3. Multiply the next corresponding A-side 24 cm by this factor to obtain 72 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 72/3=24 cm. The B perimeter would be 180 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.3333, not 3. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

51. Missing corresponding side 14: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 15, 20, 25 cm. Triangle B’s side matching the 20-cm side is 80 cm. Find the B-side matching A’s 25-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=80/20=4. Multiply the next corresponding A-side 25 cm by this factor to obtain 100 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 100/4=25 cm. The B perimeter would be 240 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.25, not 4. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

52. Missing corresponding side 15: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 3, 5, 7 cm. Triangle B’s side matching the 7-cm side is 3.5 cm. Find the B-side matching A’s 3-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=3.5/7=0.5. Multiply the next corresponding A-side 3 cm by this factor to obtain 1.5 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 1.5/0.5=3 cm. The B perimeter would be 7.5 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 2, not 0.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

53. Missing corresponding side 16: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 3, 4, 5 cm. Triangle B’s side matching the 3-cm side is 4.5 cm. Find the B-side matching A’s 4-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=4.5/3=1.5. Multiply the next corresponding A-side 4 cm by this factor to obtain 6 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 6/1.5=4 cm. The B perimeter would be 18 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.6667, not 1.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

54. Missing corresponding side 17: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 7, 24, 25 cm. Triangle B’s side matching the 24-cm side is 60 cm. Find the B-side matching A’s 25-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=60/24=2.5. Multiply the next corresponding A-side 25 cm by this factor to obtain 62.5 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 62.5/2.5=25 cm. The B perimeter would be 140 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.4, not 2.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

55. Missing corresponding side 18: exact scale factor

Try this. Triangle A has sides 10, 24, 26 cm. A similar triangle B has a side of 26/3 cm corresponding to A’s 26-cm side. Find B’s side corresponding to A’s 10-cm side. The lengths are given exactly as fractions where necessary.

Worked reasoning. The new-to-old scale factor is (26/3)/26 = 1/3. Multiply A’s 10 cm by 1/3: the required B side is 10/3 cm. Similarity is a stated fact in the question, so the single known matched pair determines its positive scale factor.

Independent check. Reverse the result: (10/3)÷(1/3)=10 cm. The new perimeter would be 20 cm compared with original 60 cm, again preserving scale factor 1/3.

56. Missing corresponding side 19: exact scale factor

Try this. Triangle A has sides 15, 20, 25 cm. A similar triangle B has a side of 20 cm corresponding to A’s 15-cm side. Find B’s side corresponding to A’s 20-cm side. The lengths are given exactly as fractions where necessary.

Worked reasoning. The new-to-old scale factor is (20)/15 = 4/3. Multiply A’s 20 cm by 4/3: the required B side is 80/3 cm. Similarity is a stated fact in the question, so the single known matched pair determines its positive scale factor.

Independent check. Reverse the result: (80/3)÷(4/3)=20 cm. The new perimeter would be 80 cm compared with original 60 cm, again preserving scale factor 4/3.

57. Missing corresponding side 20: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 3, 5, 7 cm. Triangle B’s side matching the 5-cm side is 6 cm. Find the B-side matching A’s 7-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=6/5=1.2. Multiply the next corresponding A-side 7 cm by this factor to obtain 8.4 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 8.4/1.2=7 cm. The B perimeter would be 18 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8333, not 1.2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

58. Missing corresponding side 21: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 3, 4, 5 cm. Triangle B’s side matching the 5-cm side is 8.75 cm. Find the B-side matching A’s 3-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=8.75/5=1.75. Multiply the next corresponding A-side 3 cm by this factor to obtain 5.25 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 5.25/1.75=3 cm. The B perimeter would be 21 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5714, not 1.75. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

59. Missing corresponding side 22: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 7, 24, 25 cm. Triangle B’s side matching the 7-cm side is 8.75 cm. Find the B-side matching A’s 24-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=8.75/7=1.25. Multiply the next corresponding A-side 24 cm by this factor to obtain 30 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 30/1.25=24 cm. The B perimeter would be 70 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8, not 1.25. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

60. Missing corresponding side 23: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 10, 24, 26 cm. Triangle B’s side matching the 24-cm side is 24 cm. Find the B-side matching A’s 26-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=24/24=1. Multiply the next corresponding A-side 26 cm by this factor to obtain 26 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 26/1=26 cm. The B perimeter would be 60 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 1, not 1. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

61. Missing corresponding side 24: exact scale factor

Try this. Triangle A has sides 15, 20, 25 cm. A similar triangle B has a side of 50/3 cm corresponding to A’s 25-cm side. Find B’s side corresponding to A’s 15-cm side. The lengths are given exactly as fractions where necessary.

Worked reasoning. The new-to-old scale factor is (50/3)/25 = 2/3. Multiply A’s 15 cm by 2/3: the required B side is 10 cm. Similarity is a stated fact in the question, so the single known matched pair determines its positive scale factor.

Independent check. Reverse the result: (10)÷(2/3)=15 cm. The new perimeter would be 40 cm compared with original 60 cm, again preserving scale factor 2/3.

62. Missing corresponding side 25: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 3, 5, 7 cm. Triangle B’s side matching the 3-cm side is 6 cm. Find the B-side matching A’s 5-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=6/3=2. Multiply the next corresponding A-side 5 cm by this factor to obtain 10 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 10/2=5 cm. The B perimeter would be 30 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5, not 2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

63. Missing corresponding side 26: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 3, 4, 5 cm. Triangle B’s side matching the 4-cm side is 12 cm. Find the B-side matching A’s 5-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=12/4=3. Multiply the next corresponding A-side 5 cm by this factor to obtain 15 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 15/3=5 cm. The B perimeter would be 36 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.3333, not 3. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

64. Missing corresponding side 27: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 7, 24, 25 cm. Triangle B’s side matching the 25-cm side is 100 cm. Find the B-side matching A’s 7-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=100/25=4. Multiply the next corresponding A-side 7 cm by this factor to obtain 28 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 28/4=7 cm. The B perimeter would be 224 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.25, not 4. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

65. Missing corresponding side 28: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 10, 24, 26 cm. Triangle B’s side matching the 10-cm side is 5 cm. Find the B-side matching A’s 24-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=5/10=0.5. Multiply the next corresponding A-side 24 cm by this factor to obtain 12 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 12/0.5=24 cm. The B perimeter would be 30 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 2, not 0.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

66. Missing corresponding side 29: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 15, 20, 25 cm. Triangle B’s side matching the 20-cm side is 30 cm. Find the B-side matching A’s 25-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=30/20=1.5. Multiply the next corresponding A-side 25 cm by this factor to obtain 37.5 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 37.5/1.5=25 cm. The B perimeter would be 90 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.6667, not 1.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

67. Missing corresponding side 30: derive the factor

Try this. Two triangles are stated to be similar. Triangle A’s corresponding sides are 3, 5, 7 cm. Triangle B’s side matching the 7-cm side is 17.5 cm. Find the B-side matching A’s 3-cm side and state the scale factor from A to B.

Worked reasoning. The scale factor is B/A=17.5/7=2.5. Multiply the next corresponding A-side 3 cm by this factor to obtain 7.5 cm. The statement of similarity is given, so one known matched pair is enough to find its common multiplier.

Independent check. Reverse the conversion: 7.5/2.5=3 cm. The B perimeter would be 37.5 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.4, not 2.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

68. Missing corresponding side 31: exact scale factor

Try this. Triangle A has sides 3, 4, 5 cm. A similar triangle B has a side of 1 cm corresponding to A’s 3-cm side. Find B’s side corresponding to A’s 4-cm side. The lengths are given exactly as fractions where necessary.

Worked reasoning. The new-to-old scale factor is (1)/3 = 1/3. Multiply A’s 4 cm by 1/3: the required B side is 4/3 cm. Similarity is a stated fact in the question, so the single known matched pair determines its positive scale factor.

Independent check. Reverse the result: (4/3)÷(1/3)=4 cm. The new perimeter would be 4 cm compared with original 12 cm, again preserving scale factor 1/3.

69. Rectangle aspect-ratio check 1

Try this. Rectangle A measures 3 by 4 cm and rectangle B measures 6 by 8 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 6/3=2; the matching height multiplier is 8/4=2. They agree, so the rectangles are similar. Their common factor is 2 and their four right angles already match. Their areas are 12 and 48 cm².

First possible misconception. The word ‘rectangle’ guarantees equal corresponding angles, but it does not guarantee that every pair of rectangles is similar. The side proportions matter. Comparing area alone may also hide different aspect ratios.

Independent check. The shape ratios are 3:4 and 6:8. They simplify to the same comparison; the area multiplier is 4.

Transfer. Rotate one rectangle and check whether its corresponding sides should be swapped before deciding similarity. A rotated drawing does not change intrinsic shape, but the labels must be matched consistently.

70. Rectangle aspect-ratio check 2

Try this. Rectangle A measures 3 by 5 cm and rectangle B measures 9 by 15 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 9/3=3; the matching height multiplier is 15/5=3. They agree, so the rectangles are similar. Their common factor is 3 and their four right angles already match. Their areas are 15 and 135 cm².

Independent check. The shape ratios are 3:5 and 9:15. They simplify to the same comparison; the area multiplier is 9.

71. Rectangle aspect-ratio check 3

Try this. Rectangle A measures 2 by 7 cm and rectangle B measures 4 by 14 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 4/2=2; the matching height multiplier is 14/7=2. They agree, so the rectangles are similar. Their common factor is 2 and their four right angles already match. Their areas are 14 and 56 cm².

Independent check. The shape ratios are 2:7 and 4:14. They simplify to the same comparison; the area multiplier is 4.

72. Rectangle aspect-ratio check 4

Try this. Rectangle A measures 4 by 9 cm and rectangle B measures 8 by 18 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 8/4=2; the matching height multiplier is 18/9=2. They agree, so the rectangles are similar. Their common factor is 2 and their four right angles already match. Their areas are 36 and 144 cm².

Independent check. The shape ratios are 4:9 and 8:18. They simplify to the same comparison; the area multiplier is 4.

73. Rectangle aspect-ratio check 5

Try this. Rectangle A measures 4 by 7 cm and rectangle B measures 6 by 10.5 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 6/4=1.5; the matching height multiplier is 10.5/7=1.5. They agree, so the rectangles are similar. Their common factor is 1.5 and their four right angles already match. Their areas are 28 and 63 cm².

Independent check. The shape ratios are 4:7 and 6:10.5. They simplify to the same comparison; the area multiplier is 2.25.

74. Rectangle aspect-ratio check 6

Try this. Rectangle A measures 3 by 4 cm and rectangle B measures 6 by 9 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 6/3=2; the matching height multiplier is 9/4=2.25. They differ, so the rectangles are not similar in this correspondence. A genuine rectangular similarity requires both dimensions to scale by one shared multiplier. Their areas are 12 and 54 cm².

Independent check. The shape ratios are 3:4 and 6:9. They are different; a width factor 2 cannot simultaneously explain the height factor 2.25.

75. Rectangle aspect-ratio check 7

Try this. Rectangle A measures 2 by 5 cm and rectangle B measures 4 by 11 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 4/2=2; the matching height multiplier is 11/5=2.2. They differ, so the rectangles are not similar in this correspondence. A genuine rectangular similarity requires both dimensions to scale by one shared multiplier. Their areas are 10 and 44 cm².

Independent check. The shape ratios are 2:5 and 4:11. They are different; a width factor 2 cannot simultaneously explain the height factor 2.2.

76. Rectangle aspect-ratio check 8

Try this. Rectangle A measures 3 by 7 cm and rectangle B measures 6 by 12 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 6/3=2; the matching height multiplier is 12/7=1.7143. They differ, so the rectangles are not similar in this correspondence. A genuine rectangular similarity requires both dimensions to scale by one shared multiplier. Their areas are 21 and 72 cm².

Independent check. The shape ratios are 3:7 and 6:12. They are different; a width factor 2 cannot simultaneously explain the height factor 1.7143.

77. Rectangle aspect-ratio check 9

Try this. Rectangle A measures 4 by 8 cm and rectangle B measures 10 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 10/4=2.5; the matching height multiplier is 20/8=2.5. They agree, so the rectangles are similar. Their common factor is 2.5 and their four right angles already match. Their areas are 32 and 200 cm².

Independent check. The shape ratios are 4:8 and 10:20. They simplify to the same comparison; the area multiplier is 6.25.

78. Rectangle aspect-ratio check 10

Try this. Rectangle A measures 5 by 12 cm and rectangle B measures 10 by 25 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 10/5=2; the matching height multiplier is 25/12=2.0833. They differ, so the rectangles are not similar in this correspondence. A genuine rectangular similarity requires both dimensions to scale by one shared multiplier. Their areas are 60 and 250 cm².

Independent check. The shape ratios are 5:12 and 10:25. They are different; a width factor 2 cannot simultaneously explain the height factor 2.0833.

79. Rectangle aspect-ratio check 11

Try this. Rectangle A measures 2 by 9 cm and rectangle B measures 4 by 15 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 4/2=2; the matching height multiplier is 15/9=1.6667. They differ, so the rectangles are not similar in this correspondence. A genuine rectangular similarity requires both dimensions to scale by one shared multiplier. Their areas are 18 and 60 cm².

Independent check. The shape ratios are 2:9 and 4:15. They are different; a width factor 2 cannot simultaneously explain the height factor 1.6667.

80. Rectangle aspect-ratio check 12

Try this. Rectangle A measures 3 by 9 cm and rectangle B measures 6 by 18 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 6/3=2; the matching height multiplier is 18/9=2. They agree, so the rectangles are similar. Their common factor is 2 and their four right angles already match. Their areas are 27 and 108 cm².

Independent check. The shape ratios are 3:9 and 6:18. They simplify to the same comparison; the area multiplier is 4.

81. Rectangle aspect-ratio check 13

Try this. Rectangle A measures 7 by 8 cm and rectangle B measures 14 by 16 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 14/7=2; the matching height multiplier is 16/8=2. They agree, so the rectangles are similar. Their common factor is 2 and their four right angles already match. Their areas are 56 and 224 cm².

Independent check. The shape ratios are 7:8 and 14:16. They simplify to the same comparison; the area multiplier is 4.

82. Rectangle aspect-ratio check 14

Try this. Rectangle A measures 4 by 5 cm and rectangle B measures 8 by 10 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 8/4=2; the matching height multiplier is 10/5=2. They agree, so the rectangles are similar. Their common factor is 2 and their four right angles already match. Their areas are 20 and 80 cm².

Independent check. The shape ratios are 4:5 and 8:10. They simplify to the same comparison; the area multiplier is 4.

83. Rectangle aspect-ratio check 15

Try this. Rectangle A measures 3 by 6 cm and rectangle B measures 10 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 10/3=3.3333; the matching height multiplier is 20/6=3.3333. They agree, so the rectangles are similar. Their common factor is 3.3333 and their four right angles already match. Their areas are 18 and 200 cm².

Independent check. The shape ratios are 3:6 and 10:20. They simplify to the same comparison; the area multiplier is 11.1111.

84. Rectangle aspect-ratio check 16

Try this. Rectangle A measures 3 by 8 cm and rectangle B measures 9 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 9/3=3; the matching height multiplier is 20/8=2.5. They differ, so the rectangles are not similar in this correspondence. A genuine rectangular similarity requires both dimensions to scale by one shared multiplier. Their areas are 24 and 180 cm².

Independent check. The shape ratios are 3:8 and 9:20. They are different; a width factor 3 cannot simultaneously explain the height factor 2.5.

85. Rectangle aspect-ratio check 17

Try this. Rectangle A measures 2 by 3 cm and rectangle B measures 6 by 9 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 6/2=3; the matching height multiplier is 9/3=3. They agree, so the rectangles are similar. Their common factor is 3 and their four right angles already match. Their areas are 6 and 54 cm².

Independent check. The shape ratios are 2:3 and 6:9. They simplify to the same comparison; the area multiplier is 9.

86. Rectangle aspect-ratio check 18

Try this. Rectangle A measures 5 by 7 cm and rectangle B measures 10 by 13 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 10/5=2; the matching height multiplier is 13/7=1.8571. They differ, so the rectangles are not similar in this correspondence. A genuine rectangular similarity requires both dimensions to scale by one shared multiplier. Their areas are 35 and 130 cm².

Independent check. The shape ratios are 5:7 and 10:13. They are different; a width factor 2 cannot simultaneously explain the height factor 1.8571.

87. Rectangle aspect-ratio check 19

Try this. Rectangle A measures 3 by 5 cm and rectangle B measures 12 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 12/3=4; the matching height multiplier is 20/5=4. They agree, so the rectangles are similar. Their common factor is 4 and their four right angles already match. Their areas are 15 and 240 cm².

Independent check. The shape ratios are 3:5 and 12:20. They simplify to the same comparison; the area multiplier is 16.

88. Rectangle aspect-ratio check 20

Try this. Rectangle A measures 2 by 7 cm and rectangle B measures 6 by 18 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Worked reasoning. The matching width multiplier is 6/2=3; the matching height multiplier is 18/7=2.5714. They differ, so the rectangles are not similar in this correspondence. A genuine rectangular similarity requires both dimensions to scale by one shared multiplier. Their areas are 14 and 108 cm².

Independent check. The shape ratios are 2:7 and 6:18. They are different; a width factor 3 cannot simultaneously explain the height factor 2.5714.

89. Congruent shapes need not face the same direction

Try this. A triangle is reflected in a mirror. Is its image congruent?

Worked reasoning. Yes. Reflection preserves all lengths and angles, so the image is congruent although its orientation reverses.

First possible misconception. Assuming rotated or reflected figures cannot be congruent because the drawing looks reversed.

Independent check. Trace or superimpose corresponding sides under the reflection.

Transfer. For a fresh drawing, label the corresponding vertices or sides and check both angle agreement and side-length proportions. A single shared feature may not establish similarity or congruence. If there is a length scale factor, explain why its effect on perimeter differs from its effect on area. Retest the claim with a counterexample when appropriate.

90. Congruence implies similarity

Try this. If two figures are congruent, must they be similar?

Worked reasoning. Yes. Every corresponding side length ratio is one and angles match, so they are similar at scale factor one.

First possible misconception. Treating congruence and similarity as disjoint categories.

Independent check. Every congruent copy fits the similarity definition with k=1.

91. Similarity does not imply congruence

Try this. If two triangles have equal corresponding angles, must they be congruent?

Worked reasoning. No. They can be similar at different sizes, for example 3–4–5 and 6–8–10.

First possible misconception. Confusing equal angles with equal physical lengths.

Independent check. Compare all three side ratios.

92. Two squares are similar

Try this. Are two squares of different side lengths similar?

Worked reasoning. Yes. All corresponding angles are 90° and every side scales by the same length factor.

First possible misconception. Calling different-size squares congruent.

Independent check. Their lengths differ, but the shape remains a square.

93. Two rectangles need not be similar

Try this. Is every rectangle similar to every other rectangle?

Worked reasoning. No. A 2-by-3 rectangle has ratio 2:3, unlike a 2-by-5 rectangle.

First possible misconception. All right angles are not enough for polygon similarity.

Independent check. Compare corresponding side ratios.

94. Scale factor one preserves all lengths

Try this. A figure is enlarged by scale factor 1. Is its image congruent?

Worked reasoning. Yes, for an ordinary similarity transformation with factor one, all corresponding lengths and angles remain equal.

First possible misconception. Assuming the word enlargement always means a size increase.

Independent check. Factor one represents unchanged size, possibly moved or reflected.

95. A length scale factor affects perimeter once

Try this. A triangle perimeter is 24 cm and a similar copy has scale factor 3. Find new perimeter.

Worked reasoning. It is 3×24=72 cm. Every side is tripled.

First possible misconception. Squaring the factor for perimeter would be wrong.

Independent check. The three individual sides all triple, so their sum triples.

96. An area scales with the square of a length factor

Try this. A rectangle area is 12 cm² and all lengths are doubled. Find new area.

Worked reasoning. The length factor is two, area factor is four, so new area=48 cm².

First possible misconception. Doubling area to 24 cm² treats it like perimeter.

Independent check. A 3-by-4 rectangle becomes 6-by-8 with area 48.

97. Area ratio can recover a length factor

Try this. Two similar plane figures have areas in ratio 9:25. What is the corresponding length ratio?

Worked reasoning. For positive side scales, take square roots: length ratio is 3:5.

First possible misconception. Using 9:25 directly as the side ratio changes the area again.

Independent check. Squares of 3 and 5 reconstruct 9 and 25.

98. Congruent triangles have equal area

Try this. Two triangles are congruent, but one is rotated. Compare area.

Worked reasoning. Their areas match. Rigid motions preserve size and shape.

First possible misconception. Assuming the rotated triangle fills a different area.

Independent check. Superimpose after rotation.

99. Equal area does not prove congruence

Try this. A 3-by-4 rectangle and a 2-by-6 rectangle both have area twelve. Are they congruent?

Worked reasoning. No. They have different matching side lengths and different perimeters.

First possible misconception. Treating area equality as all-shape equality.

Independent check. Compute perimeters fourteen and sixteen.

100. Equal perimeter does not prove similarity

Try this. A 3-by-7 rectangle and a 4-by-6 rectangle both have perimeter twenty. Are they similar?

Worked reasoning. No. Their aspect ratios differ, despite equal perimeters.

First possible misconception. Using a single boundary total as proof of shape proportions.

Independent check. Compare 3:7 with 4:6.

101. Corresponding order matters

Try this. Triangle ABC corresponds to DEF in that order. Which sides match?

Worked reasoning. AB with DE, BC with EF and AC with DF.

First possible misconception. Matching sides based on their position on paper instead of named vertices.

Independent check. Use vertex A↔D, B↔E, C↔F.

102. Similar triangles have equal corresponding angles

Try this. Two triangles are similar. Angle A corresponds to D and angle A=45°. What is D?

Worked reasoning. Angle D is also 45°. Similarity preserves angles.

First possible misconception. Multiplying an angle by the length scale factor.

Independent check. Angles stay equal even when side lengths change.

103. Congruent triangles share every corresponding side length

Try this. Two triangles are congruent, with side AB=7 cm corresponding to DE. Find DE.

Worked reasoning. DE is 7 cm.

First possible misconception. Multiplying by an enlargement factor other than one contradicts congruence.

Independent check. The corresponding side ratio is exactly one.

104. A triangle’s shape is fixed by its three sides up to congruence

Try this. Two valid triangles have matching three side lengths 3,4,5 in correspondence. Are they congruent?

Worked reasoning. Yes, the side-side-side congruence property applies.

First possible misconception. Assuming mirror orientation changes side lengths.

Independent check. Rotate or reflect one to superimpose.

105. Triangles with equal angle sets are similar

Try this. One triangle has angles 40°,60°,80°; another has 60°,80°,40°. Are they similar?

Worked reasoning. Yes, the corresponding angle sets match and the triangles are similar.

First possible misconception. Matching angle labels in the displayed order without relabelling corresponding corners.

Independent check. Angle-angle similarity suffices for Euclidean triangles.

106. Angle equality alone is not enough for general quadrilaterals

Try this. Two rectangles have four right angles but different aspect ratios. Are they necessarily similar?

Worked reasoning. No. Side proportions must agree as well.

First possible misconception. Importing the triangle angle-angle shortcut into every polygon.

Independent check. Compare 2-by-3 and 2-by-5 rectangles.

107. A scale factor can be a fraction

Try this. A triangle side 12 cm becomes 9 cm in a similar smaller copy. Find scale factor.

Worked reasoning. k=9/12=3/4, less than one.

First possible misconception. Assuming an enlargement or similarity scale must exceed one.

Independent check. New side/old side=0.75.

108. Reverse a reduction

Try this. A reduced copy has side 6 cm at factor 1/2. Find original side.

Worked reasoning. Original=6/(1/2)=12 cm.

First possible misconception. Multiplying by 1/2 again reduces rather than reverses.

Independent check. 12×1/2=6.

109. Similar figures need compatible units

Try this. One matching side is 40 cm; another is 1.2 m. Find scale from first to second.

Worked reasoning. Convert 1.2 m to 120 cm. Factor=120/40=3.

First possible misconception. Comparing 40:1.2 before common units creates a false ratio.

Independent check. 40×3=120 cm=1.2 m.

110. A diagonal scales with every length

Try this. A 3-by-4 rectangle becomes 6-by-8. Compare diagonals.

Worked reasoning. Diagonals scale from 5 cm to 10 cm: factor two.

First possible misconception. Assuming diagonals can scale differently from sides in a similar figure.

Independent check. Pythagoras gives √(3²+4²)=5 and √(6²+8²)=10.

111. Changing only one side does not preserve shape

Try this. A 3-by-4 rectangle becomes 6-by-4. Are they similar?

Worked reasoning. No. Width doubled while height stayed constant, so the aspect ratio changed.

First possible misconception. Declaring similarity because one pair of sides has the right factor.

Independent check. Check 6/3=2 but 4/4=1.

112. One valid matched side pair does not prove similarity

Try this. Two arbitrary triangles have a 5-cm side each. Does that establish similarity?

Worked reasoning. No. Another two corresponding side ratios or angle information is needed.

First possible misconception. Using one matching feature as a complete proof.

Independent check. Two triangles could share one side yet have different other sides.

113. Scalability in maps

Try this. A fictional scale drawing uses 1 cm to 50 cm. A model shape side is 6 cm on drawing. What is its real length?

Worked reasoning. 6×50=300 cm, or 3 m.

First possible misconception. Treating 1:50 as 1 cm to 50 m changes the unit by a hundredfold.

Independent check. 300/50=6 drawing centimetres.

114. A scaled area on a plan

Try this. A flat shape drawn at 1:100 has drawing area 4 cm². What ideal actual area in m²?

Worked reasoning. Area factor is 100²=10,000; real area 40,000 cm²=4 m².

First possible misconception. Multiplying by only 100 for area ignores two dimensions.

Independent check. A 2-by-2 cm square corresponds to 2-by-2 m, area 4 m².

115. Area ratio and perimeter ratio differ

Try this. Similar figures have length scale 3/2. State perimeter and area scale factors.

Worked reasoning. Perimeter factor 3/2; area factor (3/2)²=9/4.

First possible misconception. Applying 3/2 to area without squaring.

Independent check. Compare a square of side two with side three: areas four and nine.

116. Similarity preserves angles but not necessarily area

Try this. A triangle is enlarged by scale factor four. What happens to one 70° angle and its area?

Worked reasoning. Angle stays 70°; area becomes sixteen times its original.

First possible misconception. Multiplying angle by four would give an unsupported 280° corner.

Independent check. Similarity is an angle-preserving transformation with length scaling.

117. Some similar figures are congruent

Try this. Can similar figures also be congruent?

Worked reasoning. Yes, if their positive scale factor is one.

First possible misconception. Answering ‘never’ confuses broader and narrower shape categories.

Independent check. Identical-sized copies satisfy both definitions.

118. Congruence does not require the same drawing position

Try this. Two identical triangles are translated horizontally. Are they still congruent?

Worked reasoning. Yes. Translation changes position but no length or angle.

First possible misconception. Demanding identical coordinates for matching figures.

Independent check. Rigid motion gives an exact overlap after moving one.

119. Area factor works for triangles too

Try this. A triangle has area 10 cm² and a similar copy has length factor 2.5. Find new area.

Worked reasoning. Area factor 2.5²=6.25. New area=62.5 cm².

First possible misconception. Using a linear factor of 2.5 for area gives 25, too small.

Independent check. Compare base and perpendicular height, each multiplied by 2.5.

120. The shortest side should match the shortest side

Try this. Two similar triangles have side lists 4,6,9 and 8,12,18. Which sides correspond naturally?

Worked reasoning. 4↔8, 6↔12, 9↔18; all ratios two.

First possible misconception. Matching four with eighteen creates inconsistent ratios.

Independent check. Sort sides or use supplied vertex correspondence.

121. An image resized nonuniformly is distorted

Try this. A rectangle image is stretched twice as wide but unchanged in height. Is the new image similar?

Worked reasoning. No, except in degenerate cases. The aspect ratio changes, so it is a distortion rather than a uniform similarity.

First possible misconception. Calling any bigger picture a proportional enlargement.

Independent check. Compare horizontal and vertical scaling separately.

122. Angles of similar quadrilaterals

Try this. Two quadrilaterals are given similar, and a corresponding angle in one is 105°. Find the second.

Worked reasoning. It is 105°, because similarity preserves corresponding angle size.

First possible misconception. Scaling angles using a side factor.

Independent check. Identify corresponding vertices and apply equality.

123. Congruent isn’t the same as equal perimeters alone

Try this. Two different rectangles both have 20-cm perimeter. Must they be congruent?

Worked reasoning. No. 3-by-7 and 4-by-6 both have perimeter twenty but different side lengths.

First possible misconception. Boundary equality is not an entire shape description.

Independent check. A congruence claim needs matched geometry.

124. Area formula can expose a mismatch

Try this. A right triangle with legs three and four is compared to one with legs six and eight. What is their area ratio?

Worked reasoning. Areas are six and twenty-four square units, so ratio 1:4; length ratio 1:2.

First possible misconception. Using a length ratio 1:2 directly as area ratio.

Independent check. Both perpendicular legs double, giving four times the area.

125. A scale of one-third reverses with three

Try this. A side 15 cm becomes 5 cm. State downscale and reverse factors.

Worked reasoning. From original to copy k=1/3; from copy to original factor three.

First possible misconception. Using 1/3 in both directions.

Independent check. 5×3=15 and 15×1/3=5.

Four learning gaps with different repairs

The learner assumes similar means same size

When a 3–4–5 triangle is compared with a 6–8–10 triangle, the angles and proportions agree but physical lengths differ. They are similar, not congruent. Ask the student to state which extra condition would make them congruent: length scale factor one.

The wrong side pairs are compared

If AB corresponds to DE and AC corresponds to DF, calculating AB/DF can produce a meaningless ratio. Mark corresponding vertices before any division. Have the student show that all correct ratios equal one common factor. The repair is matching geometry, not arithmetic speed.

Area is scaled by the length factor instead of its square

Doubling both sides of a rectangle produces four times the area, not twice. Draw two-dimensional unit-square grids and count the growth in both directions. Then change the factor to three and predict an area factor nine. That transfer reveals whether the rule was understood.

A single shared angle is treated as complete proof

Two arbitrary triangles may share one angle yet have different other angles and shapes. The pupil needs sufficient similarity evidence, such as two matching angles for triangles or all corresponding side ratios. A single matching side is also insufficient. Learn what evidence the statement actually provides before using a scale factor.

How a three-pupil G1 Mathematics lesson can work

The immutable eduKateSG Clementi three-pupil Maths reference describes 90-minute tutorials near Sixth Avenue MRT. A differentiated similarity lesson could have one pupil match corresponding vertices, another calculate scale factors and a third explain area/perimeter changes and false claims. This is an illustrative adaptation; current group openings and class suitability require direct confirmation.

10 minutes: retrieve equal angles, parallel marks and common ratios.
20 minutes: compare a congruent and a scaled triangle with labelled correspondence.
25 minutes: calculate missing sides and check matching ratios independently.
15 minutes: compare perimeter versus area scaling and rectangular aspect ratios.
10 minutes: solve an unseen side correspondence or counterexample question.
10 minutes: explain the first mismatch and set a later retest.

Five short practice sessions

Day 1: identify congruent copies under translation, rotation and reflection.
Day 2: match corresponding sides in two named similar triangles.
Day 3: find new side lengths under whole-number and fractional scale factors.
Day 4: compare perimeter and area factors without memorising the wrong one.
Day 5: find a rectangle counterexample that shows identical angles alone are not sufficient for similarity.

Subject-level and examination-year accuracy

Frequently asked questions

Are congruent shapes also similar?

Yes. Their similarity scale factor is one and all matching measurements agree.

Are all similar shapes congruent?

No. Similar shapes may differ in size; only those with scale factor one are congruent.

Does rotating a shape stop it being congruent?

No. Rotation preserves length and angle. So do translation and reflection.

Why is area factor k²?

Area has two dimensions, both scaling by k. The combined factor is k×k=k².

Are all rectangles similar?

No. Their corresponding side ratios must agree; rectangles can have different aspect ratios.

What should a tutor check beyond the final side answer?

The matched vertex order, scale direction, uniform side ratios and whether the conclusion is similarity or congruence.

Official curriculum and reader notes

Bring one incorrect matching-side diagram to eduKate Singapore and ask how a tutor would teach corresponding sides and scale factors before expanding to harder geometry.

One-minute diagnostic
What congruent and similar mean
Find corresponding sides
Length, perimeter and area factors
Original worked shape clinics
Four common mistakes
Three-pupil tutorial model
Home practice
Frequently asked questions
Official sources

The one-minute comparison

Similarity preserves shape, congruence preserves size too

Congruence tolerates movement and reflection

Similarity requires one multiplier for matching lengths

Match corresponding vertices before dividing

Why an area does not scale like a side length

125 original correspondence and scale-factor clinics

1. Similar-triangle comparison 1: factor 2/1

2. Similar-triangle comparison 2: factor 3/1

3. Similar-triangle comparison 3: factor 4/1

4. Similar-triangle comparison 4: factor 1/2

5. Similar-triangle comparison 5: factor 3/2

6. Similar-triangle comparison 6: factor 5/2

7. Similar-triangle comparison 7: factor 1/3

8. Similar-triangle comparison 8: factor 4/3

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 4/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

9. Similar-triangle comparison 9: factor 6/5

10. Similar-triangle comparison 10: factor 7/4

11. Similar-triangle comparison 11: factor 5/4

12. Similar-triangle comparison 12: factor 1/1

13. Similar-triangle comparison 13: factor 2/3

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 2/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

14. Similar-triangle comparison 14: factor 2/1

15. Similar-triangle comparison 15: factor 3/1

16. Similar-triangle comparison 16: factor 4/1

17. Similar-triangle comparison 17: factor 1/2

18. Similar-triangle comparison 18: factor 3/2

19. Similar-triangle comparison 19: factor 5/2

20. Similar-triangle comparison 20: factor 1/3

21. Similar-triangle comparison 21: factor 4/3

22. Similar-triangle comparison 22: factor 6/5

23. Similar-triangle comparison 23: factor 7/4

24. Similar-triangle comparison 24: factor 5/4

25. Similar-triangle comparison 25: factor 1/1

26. Similar-triangle comparison 26: factor 2/3

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 2/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

27. Similar-triangle comparison 27: factor 2/1

28. Similar-triangle comparison 28: factor 3/1

29. Similar-triangle comparison 29: factor 4/1

30. Similar-triangle comparison 30: factor 1/2

31. Similar-triangle comparison 31: factor 3/2

32. Similar-triangle comparison 32: factor 5/2

33. Similar-triangle comparison 33: factor 1/3

34. Similar-triangle comparison 34: factor 4/3

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 4/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

35. Similar-triangle comparison 35: factor 6/5

36. Similar-triangle comparison 36: factor 7/4

37. Similar-triangle comparison 37: factor 5/4

38. Missing corresponding side 1: derive the factor

Independent check. Reverse the conversion: 16/4=4 cm. The B perimeter would be 48 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.25, not 4. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

39. Missing corresponding side 2: derive the factor

Independent check. Reverse the conversion: 12.5/0.5=25 cm. The B perimeter would be 28 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 2, not 0.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

40. Missing corresponding side 3: derive the factor

Independent check. Reverse the conversion: 15/1.5=10 cm. The B perimeter would be 90 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.6667, not 1.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

41. Missing corresponding side 4: derive the factor

Independent check. Reverse the conversion: 50/2.5=20 cm. The B perimeter would be 150 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.4, not 2.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

42. Missing corresponding side 5: exact scale factor

Independent check. Reverse the result: (7/3)÷(1/3)=7 cm. The new perimeter would be 5 cm compared with original 15 cm, again preserving scale factor 1/3.

43. Missing corresponding side 6: exact scale factor

Independent check. Reverse the result: (4)÷(4/3)=3 cm. The new perimeter would be 16 cm compared with original 12 cm, again preserving scale factor 4/3.

44. Missing corresponding side 7: derive the factor

Independent check. Reverse the conversion: 28.8/1.2=24 cm. The B perimeter would be 67.2 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8333, not 1.2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

45. Missing corresponding side 8: derive the factor

Independent check. Reverse the conversion: 45.5/1.75=26 cm. The B perimeter would be 105 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5714, not 1.75. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

46. Missing corresponding side 9: derive the factor

Independent check. Reverse the conversion: 18.75/1.25=15 cm. The B perimeter would be 75 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8, not 1.25. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

47. Missing corresponding side 10: derive the factor

Independent check. Reverse the conversion: 5/1=5 cm. The B perimeter would be 15 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 1, not 1. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

48. Missing corresponding side 11: exact scale factor

Independent check. Reverse the result: (10/3)÷(2/3)=5 cm. The new perimeter would be 8 cm compared with original 12 cm, again preserving scale factor 2/3.

49. Missing corresponding side 12: derive the factor

Independent check. Reverse the conversion: 14/2=7 cm. The B perimeter would be 112 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5, not 2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

50. Missing corresponding side 13: derive the factor

Independent check. Reverse the conversion: 72/3=24 cm. The B perimeter would be 180 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.3333, not 3. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

51. Missing corresponding side 14: derive the factor

Independent check. Reverse the conversion: 100/4=25 cm. The B perimeter would be 240 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.25, not 4. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

52. Missing corresponding side 15: derive the factor

Independent check. Reverse the conversion: 1.5/0.5=3 cm. The B perimeter would be 7.5 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 2, not 0.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

53. Missing corresponding side 16: derive the factor

Independent check. Reverse the conversion: 6/1.5=4 cm. The B perimeter would be 18 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.6667, not 1.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

54. Missing corresponding side 17: derive the factor

Independent check. Reverse the conversion: 62.5/2.5=25 cm. The B perimeter would be 140 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.4, not 2.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

55. Missing corresponding side 18: exact scale factor

Independent check. Reverse the result: (10/3)÷(1/3)=10 cm. The new perimeter would be 20 cm compared with original 60 cm, again preserving scale factor 1/3.

56. Missing corresponding side 19: exact scale factor

Independent check. Reverse the result: (80/3)÷(4/3)=20 cm. The new perimeter would be 80 cm compared with original 60 cm, again preserving scale factor 4/3.

57. Missing corresponding side 20: derive the factor

Independent check. Reverse the conversion: 8.4/1.2=7 cm. The B perimeter would be 18 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8333, not 1.2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

58. Missing corresponding side 21: derive the factor

Independent check. Reverse the conversion: 5.25/1.75=3 cm. The B perimeter would be 21 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5714, not 1.75. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

59. Missing corresponding side 22: derive the factor

Independent check. Reverse the conversion: 30/1.25=24 cm. The B perimeter would be 70 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8, not 1.25. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

60. Missing corresponding side 23: derive the factor

Independent check. Reverse the conversion: 26/1=26 cm. The B perimeter would be 60 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 1, not 1. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

61. Missing corresponding side 24: exact scale factor

Independent check. Reverse the result: (10)÷(2/3)=15 cm. The new perimeter would be 40 cm compared with original 60 cm, again preserving scale factor 2/3.

62. Missing corresponding side 25: derive the factor

Independent check. Reverse the conversion: 10/2=5 cm. The B perimeter would be 30 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5, not 2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

63. Missing corresponding side 26: derive the factor

Independent check. Reverse the conversion: 15/3=5 cm. The B perimeter would be 36 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.3333, not 3. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

64. Missing corresponding side 27: derive the factor

Independent check. Reverse the conversion: 28/4=7 cm. The B perimeter would be 224 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.25, not 4. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

65. Missing corresponding side 28: derive the factor

Independent check. Reverse the conversion: 12/0.5=24 cm. The B perimeter would be 30 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 2, not 0.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

66. Missing corresponding side 29: derive the factor

Independent check. Reverse the conversion: 37.5/1.5=25 cm. The B perimeter would be 90 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.6667, not 1.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

67. Missing corresponding side 30: derive the factor

Independent check. Reverse the conversion: 7.5/2.5=3 cm. The B perimeter would be 37.5 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.4, not 2.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

68. Missing corresponding side 31: exact scale factor

Independent check. Reverse the result: (4/3)÷(1/3)=4 cm. The new perimeter would be 4 cm compared with original 12 cm, again preserving scale factor 1/3.

69. Rectangle aspect-ratio check 1

Try this. Rectangle A measures 3 by 4 cm and rectangle B measures 6 by 8 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:4 and 6:8. They simplify to the same comparison; the area multiplier is 4.

70. Rectangle aspect-ratio check 2

Try this. Rectangle A measures 3 by 5 cm and rectangle B measures 9 by 15 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:5 and 9:15. They simplify to the same comparison; the area multiplier is 9.

71. Rectangle aspect-ratio check 3

Try this. Rectangle A measures 2 by 7 cm and rectangle B measures 4 by 14 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 2:7 and 4:14. They simplify to the same comparison; the area multiplier is 4.

72. Rectangle aspect-ratio check 4

Try this. Rectangle A measures 4 by 9 cm and rectangle B measures 8 by 18 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 4:9 and 8:18. They simplify to the same comparison; the area multiplier is 4.

73. Rectangle aspect-ratio check 5

Try this. Rectangle A measures 4 by 7 cm and rectangle B measures 6 by 10.5 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 4:7 and 6:10.5. They simplify to the same comparison; the area multiplier is 2.25.

74. Rectangle aspect-ratio check 6

Try this. Rectangle A measures 3 by 4 cm and rectangle B measures 6 by 9 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:4 and 6:9. They are different; a width factor 2 cannot simultaneously explain the height factor 2.25.

75. Rectangle aspect-ratio check 7

Try this. Rectangle A measures 2 by 5 cm and rectangle B measures 4 by 11 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 2:5 and 4:11. They are different; a width factor 2 cannot simultaneously explain the height factor 2.2.

76. Rectangle aspect-ratio check 8

Try this. Rectangle A measures 3 by 7 cm and rectangle B measures 6 by 12 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:7 and 6:12. They are different; a width factor 2 cannot simultaneously explain the height factor 1.7143.

77. Rectangle aspect-ratio check 9

Try this. Rectangle A measures 4 by 8 cm and rectangle B measures 10 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 4:8 and 10:20. They simplify to the same comparison; the area multiplier is 6.25.

78. Rectangle aspect-ratio check 10

Try this. Rectangle A measures 5 by 12 cm and rectangle B measures 10 by 25 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 5:12 and 10:25. They are different; a width factor 2 cannot simultaneously explain the height factor 2.0833.

79. Rectangle aspect-ratio check 11

Try this. Rectangle A measures 2 by 9 cm and rectangle B measures 4 by 15 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 2:9 and 4:15. They are different; a width factor 2 cannot simultaneously explain the height factor 1.6667.

80. Rectangle aspect-ratio check 12

Try this. Rectangle A measures 3 by 9 cm and rectangle B measures 6 by 18 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:9 and 6:18. They simplify to the same comparison; the area multiplier is 4.

81. Rectangle aspect-ratio check 13

Try this. Rectangle A measures 7 by 8 cm and rectangle B measures 14 by 16 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 7:8 and 14:16. They simplify to the same comparison; the area multiplier is 4.

82. Rectangle aspect-ratio check 14

Try this. Rectangle A measures 4 by 5 cm and rectangle B measures 8 by 10 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 4:5 and 8:10. They simplify to the same comparison; the area multiplier is 4.

83. Rectangle aspect-ratio check 15

Try this. Rectangle A measures 3 by 6 cm and rectangle B measures 10 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:6 and 10:20. They simplify to the same comparison; the area multiplier is 11.1111.

84. Rectangle aspect-ratio check 16

Try this. Rectangle A measures 3 by 8 cm and rectangle B measures 9 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:8 and 9:20. They are different; a width factor 3 cannot simultaneously explain the height factor 2.5.

85. Rectangle aspect-ratio check 17

Try this. Rectangle A measures 2 by 3 cm and rectangle B measures 6 by 9 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 2:3 and 6:9. They simplify to the same comparison; the area multiplier is 9.

86. Rectangle aspect-ratio check 18

Try this. Rectangle A measures 5 by 7 cm and rectangle B measures 10 by 13 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 5:7 and 10:13. They are different; a width factor 2 cannot simultaneously explain the height factor 1.8571.

87. Rectangle aspect-ratio check 19

Try this. Rectangle A measures 3 by 5 cm and rectangle B measures 12 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:5 and 12:20. They simplify to the same comparison; the area multiplier is 16.

88. Rectangle aspect-ratio check 20

Try this. Rectangle A measures 2 by 7 cm and rectangle B measures 6 by 18 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 2:7 and 6:18. They are different; a width factor 3 cannot simultaneously explain the height factor 2.5714.

89. Congruent shapes need not face the same direction

Try this. A triangle is reflected in a mirror. Is its image congruent?

Worked reasoning. Yes. Reflection preserves all lengths and angles, so the image is congruent although its orientation reverses.

First possible misconception. Assuming rotated or reflected figures cannot be congruent because the drawing looks reversed.

Independent check. Trace or superimpose corresponding sides under the reflection.

90. Congruence implies similarity

Try this. If two figures are congruent, must they be similar?

Worked reasoning. Yes. Every corresponding side length ratio is one and angles match, so they are similar at scale factor one.

First possible misconception. Treating congruence and similarity as disjoint categories.

Independent check. Every congruent copy fits the similarity definition with k=1.

91. Similarity does not imply congruence

Try this. If two triangles have equal corresponding angles, must they be congruent?

Worked reasoning. No. They can be similar at different sizes, for example 3–4–5 and 6–8–10.

First possible misconception. Confusing equal angles with equal physical lengths.

Independent check. Compare all three side ratios.

92. Two squares are similar

Try this. Are two squares of different side lengths similar?

Worked reasoning. Yes. All corresponding angles are 90° and every side scales by the same length factor.

First possible misconception. Calling different-size squares congruent.

Independent check. Their lengths differ, but the shape remains a square.

93. Two rectangles need not be similar

Try this. Is every rectangle similar to every other rectangle?

Worked reasoning. No. A 2-by-3 rectangle has ratio 2:3, unlike a 2-by-5 rectangle.

First possible misconception. All right angles are not enough for polygon similarity.

Independent check. Compare corresponding side ratios.

94. Scale factor one preserves all lengths

Try this. A figure is enlarged by scale factor 1. Is its image congruent?

Worked reasoning. Yes, for an ordinary similarity transformation with factor one, all corresponding lengths and angles remain equal.

First possible misconception. Assuming the word enlargement always means a size increase.

Independent check. Factor one represents unchanged size, possibly moved or reflected.

95. A length scale factor affects perimeter once

Try this. A triangle perimeter is 24 cm and a similar copy has scale factor 3. Find new perimeter.

Worked reasoning. It is 3×24=72 cm. Every side is tripled.

First possible misconception. Squaring the factor for perimeter would be wrong.

Independent check. The three individual sides all triple, so their sum triples.

96. An area scales with the square of a length factor

Try this. A rectangle area is 12 cm² and all lengths are doubled. Find new area.

Worked reasoning. The length factor is two, area factor is four, so new area=48 cm².

First possible misconception. Doubling area to 24 cm² treats it like perimeter.

Independent check. A 3-by-4 rectangle becomes 6-by-8 with area 48.

97. Area ratio can recover a length factor

Try this. Two similar plane figures have areas in ratio 9:25. What is the corresponding length ratio?

Worked reasoning. For positive side scales, take square roots: length ratio is 3:5.

First possible misconception. Using 9:25 directly as the side ratio changes the area again.

Independent check. Squares of 3 and 5 reconstruct 9 and 25.

98. Congruent triangles have equal area

Try this. Two triangles are congruent, but one is rotated. Compare area.

Worked reasoning. Their areas match. Rigid motions preserve size and shape.

First possible misconception. Assuming the rotated triangle fills a different area.

Independent check. Superimpose after rotation.

99. Equal area does not prove congruence

Try this. A 3-by-4 rectangle and a 2-by-6 rectangle both have area twelve. Are they congruent?

Worked reasoning. No. They have different matching side lengths and different perimeters.

First possible misconception. Treating area equality as all-shape equality.

Independent check. Compute perimeters fourteen and sixteen.

100. Equal perimeter does not prove similarity

Try this. A 3-by-7 rectangle and a 4-by-6 rectangle both have perimeter twenty. Are they similar?

Worked reasoning. No. Their aspect ratios differ, despite equal perimeters.

First possible misconception. Using a single boundary total as proof of shape proportions.

Independent check. Compare 3:7 with 4:6.

101. Corresponding order matters

Try this. Triangle ABC corresponds to DEF in that order. Which sides match?

Worked reasoning. AB with DE, BC with EF and AC with DF.

First possible misconception. Matching sides based on their position on paper instead of named vertices.

Independent check. Use vertex A↔D, B↔E, C↔F.

102. Similar triangles have equal corresponding angles

Try this. Two triangles are similar. Angle A corresponds to D and angle A=45°. What is D?

Worked reasoning. Angle D is also 45°. Similarity preserves angles.

First possible misconception. Multiplying an angle by the length scale factor.

Independent check. Angles stay equal even when side lengths change.

103. Congruent triangles share every corresponding side length

Try this. Two triangles are congruent, with side AB=7 cm corresponding to DE. Find DE.

Worked reasoning. DE is 7 cm.

First possible misconception. Multiplying by an enlargement factor other than one contradicts congruence.

Independent check. The corresponding side ratio is exactly one.

104. A triangle’s shape is fixed by its three sides up to congruence

Try this. Two valid triangles have matching three side lengths 3,4,5 in correspondence. Are they congruent?

Worked reasoning. Yes, the side-side-side congruence property applies.

First possible misconception. Assuming mirror orientation changes side lengths.

Independent check. Rotate or reflect one to superimpose.

105. Triangles with equal angle sets are similar

Try this. One triangle has angles 40°,60°,80°; another has 60°,80°,40°. Are they similar?

Worked reasoning. Yes, the corresponding angle sets match and the triangles are similar.

First possible misconception. Matching angle labels in the displayed order without relabelling corresponding corners.

Independent check. Angle-angle similarity suffices for Euclidean triangles.

106. Angle equality alone is not enough for general quadrilaterals

Try this. Two rectangles have four right angles but different aspect ratios. Are they necessarily similar?

Worked reasoning. No. Side proportions must agree as well.

First possible misconception. Importing the triangle angle-angle shortcut into every polygon.

Independent check. Compare 2-by-3 and 2-by-5 rectangles.

107. A scale factor can be a fraction

Try this. A triangle side 12 cm becomes 9 cm in a similar smaller copy. Find scale factor.

Worked reasoning. k=9/12=3/4, less than one.

First possible misconception. Assuming an enlargement or similarity scale must exceed one.

Independent check. New side/old side=0.75.

108. Reverse a reduction

Try this. A reduced copy has side 6 cm at factor 1/2. Find original side.

Worked reasoning. Original=6/(1/2)=12 cm.

First possible misconception. Multiplying by 1/2 again reduces rather than reverses.

Independent check. 12×1/2=6.

109. Similar figures need compatible units

Try this. One matching side is 40 cm; another is 1.2 m. Find scale from first to second.

Worked reasoning. Convert 1.2 m to 120 cm. Factor=120/40=3.

First possible misconception. Comparing 40:1.2 before common units creates a false ratio.

Independent check. 40×3=120 cm=1.2 m.

110. A diagonal scales with every length

Try this. A 3-by-4 rectangle becomes 6-by-8. Compare diagonals.

Worked reasoning. Diagonals scale from 5 cm to 10 cm: factor two.

First possible misconception. Assuming diagonals can scale differently from sides in a similar figure.

Independent check. Pythagoras gives √(3²+4²)=5 and √(6²+8²)=10.

111. Changing only one side does not preserve shape

Try this. A 3-by-4 rectangle becomes 6-by-4. Are they similar?

Worked reasoning. No. Width doubled while height stayed constant, so the aspect ratio changed.

First possible misconception. Declaring similarity because one pair of sides has the right factor.

Independent check. Check 6/3=2 but 4/4=1.

112. One valid matched side pair does not prove similarity

Try this. Two arbitrary triangles have a 5-cm side each. Does that establish similarity?

Worked reasoning. No. Another two corresponding side ratios or angle information is needed.

First possible misconception. Using one matching feature as a complete proof.

Independent check. Two triangles could share one side yet have different other sides.

113. Scalability in maps

Try this. A fictional scale drawing uses 1 cm to 50 cm. A model shape side is 6 cm on drawing. What is its real length?

Worked reasoning. 6×50=300 cm, or 3 m.

First possible misconception. Treating 1:50 as 1 cm to 50 m changes the unit by a hundredfold.

Independent check. 300/50=6 drawing centimetres.

114. A scaled area on a plan

Try this. A flat shape drawn at 1:100 has drawing area 4 cm². What ideal actual area in m²?

Worked reasoning. Area factor is 100²=10,000; real area 40,000 cm²=4 m².

First possible misconception. Multiplying by only 100 for area ignores two dimensions.

Independent check. A 2-by-2 cm square corresponds to 2-by-2 m, area 4 m².

115. Area ratio and perimeter ratio differ

Try this. Similar figures have length scale 3/2. State perimeter and area scale factors.

Worked reasoning. Perimeter factor 3/2; area factor (3/2)²=9/4.

First possible misconception. Applying 3/2 to area without squaring.

Independent check. Compare a square of side two with side three: areas four and nine.

116. Similarity preserves angles but not necessarily area

Try this. A triangle is enlarged by scale factor four. What happens to one 70° angle and its area?

Worked reasoning. Angle stays 70°; area becomes sixteen times its original.

First possible misconception. Multiplying angle by four would give an unsupported 280° corner.

Independent check. Similarity is an angle-preserving transformation with length scaling.

117. Some similar figures are congruent

Try this. Can similar figures also be congruent?

Worked reasoning. Yes, if their positive scale factor is one.

First possible misconception. Answering ‘never’ confuses broader and narrower shape categories.

Independent check. Identical-sized copies satisfy both definitions.

118. Congruence does not require the same drawing position

Try this. Two identical triangles are translated horizontally. Are they still congruent?

Worked reasoning. Yes. Translation changes position but no length or angle.

First possible misconception. Demanding identical coordinates for matching figures.

Independent check. Rigid motion gives an exact overlap after moving one.

119. Area factor works for triangles too

Try this. A triangle has area 10 cm² and a similar copy has length factor 2.5. Find new area.

Worked reasoning. Area factor 2.5²=6.25. New area=62.5 cm².

First possible misconception. Using a linear factor of 2.5 for area gives 25, too small.

Independent check. Compare base and perpendicular height, each multiplied by 2.5.

120. The shortest side should match the shortest side

Try this. Two similar triangles have side lists 4,6,9 and 8,12,18. Which sides correspond naturally?

Worked reasoning. 4↔8, 6↔12, 9↔18; all ratios two.

First possible misconception. Matching four with eighteen creates inconsistent ratios.

Independent check. Sort sides or use supplied vertex correspondence.

121. An image resized nonuniformly is distorted

Try this. A rectangle image is stretched twice as wide but unchanged in height. Is the new image similar?

Worked reasoning. No, except in degenerate cases. The aspect ratio changes, so it is a distortion rather than a uniform similarity.

First possible misconception. Calling any bigger picture a proportional enlargement.

Independent check. Compare horizontal and vertical scaling separately.

122. Angles of similar quadrilaterals

Try this. Two quadrilaterals are given similar, and a corresponding angle in one is 105°. Find the second.

Worked reasoning. It is 105°, because similarity preserves corresponding angle size.

First possible misconception. Scaling angles using a side factor.

Independent check. Identify corresponding vertices and apply equality.

123. Congruent isn’t the same as equal perimeters alone

Try this. Two different rectangles both have 20-cm perimeter. Must they be congruent?

Worked reasoning. No. 3-by-7 and 4-by-6 both have perimeter twenty but different side lengths.

First possible misconception. Boundary equality is not an entire shape description.

Independent check. A congruence claim needs matched geometry.

124. Area formula can expose a mismatch

Try this. A right triangle with legs three and four is compared to one with legs six and eight. What is their area ratio?

Worked reasoning. Areas are six and twenty-four square units, so ratio 1:4; length ratio 1:2.

First possible misconception. Using a length ratio 1:2 directly as area ratio.

Independent check. Both perpendicular legs double, giving four times the area.

125. A scale of one-third reverses with three

Try this. A side 15 cm becomes 5 cm. State downscale and reverse factors.

Worked reasoning. From original to copy k=1/3; from copy to original factor three.

First possible misconception. Using 1/3 in both directions.

Independent check. 5×3=15 and 15×1/3=5.

Four learning gaps with different repairs

The learner assumes similar means same size

The wrong side pairs are compared

Area is scaled by the length factor instead of its square

A single shared angle is treated as complete proof

How a three-pupil G1 Mathematics lesson can work

10 minutes: retrieve equal angles, parallel marks and common ratios.
20 minutes: compare a congruent and a scaled triangle with labelled correspondence.
25 minutes: calculate missing sides and check matching ratios independently.
15 minutes: compare perimeter versus area scaling and rectangular aspect ratios.
10 minutes: solve an unseen side correspondence or counterexample question.
10 minutes: explain the first mismatch and set a later retest.

Five short practice sessions

Day 1: identify congruent copies under translation, rotation and reflection.
Day 2: match corresponding sides in two named similar triangles.
Day 3: find new side lengths under whole-number and fractional scale factors.
Day 4: compare perimeter and area factors without memorising the wrong one.
Day 5: find a rectangle counterexample that shows identical angles alone are not sufficient for similarity.

Subject-level and examination-year accuracy

Frequently asked questions

Are congruent shapes also similar?

Yes. Their similarity scale factor is one and all matching measurements agree.

Are all similar shapes congruent?

No. Similar shapes may differ in size; only those with scale factor one are congruent.

Does rotating a shape stop it being congruent?

No. Rotation preserves length and angle. So do translation and reflection.

Why is area factor k²?

Area has two dimensions, both scaling by k. The combined factor is k×k=k².

Are all rectangles similar?

No. Their corresponding side ratios must agree; rectangles can have different aspect ratios.

What should a tutor check beyond the final side answer?

The matched vertex order, scale direction, uniform side ratios and whether the conclusion is similarity or congruence.

Official curriculum and reader notes

Bring one incorrect matching-side diagram to eduKate Singapore and ask how a tutor would teach corresponding sides and scale factors before expanding to harder geometry.

Did you know? A square and a larger square are similar even though they are not congruent. And two very different-looking triangles may be congruent if one has simply been turned over and rotated. This is why the words similar and congruent should be learned as mathematical properties, not as descriptions of whether two drawings look alike.

Quick answer: congruent versus similar shapes

Congruent figures match exactly in shape and size: corresponding sides and angles are equal, even after moving, rotating or reflecting the figure. Similar figures have equal corresponding angles and proportional corresponding lengths; their size may differ by a single positive scale factor. All congruent figures are similar at factor one, but similar figures enlarged or reduced by a non-unit factor are not congruent. For a scale factor k, perimeters multiply by k and areas by k².

This guide is for parents exploring Secondary 2 G1 Mathematics tuition in Bukit Timah. MOE’s G1 Secondary Two Mathematics syllabus includes congruent and similar figures and properties of similar triangles and quadrilaterals, including equal corresponding angles and proportional corresponding sides. The specific focus here is matching sides and understanding length and area scale, rather than a general introduction to every geometry topic.

The one-minute comparison

Similarity preserves shape, congruence preserves size too

Congruence tolerates movement and reflection

Similarity requires one multiplier for matching lengths

Match corresponding vertices before dividing

Why an area does not scale like a side length

125 original correspondence and scale-factor clinics

1. Similar-triangle comparison 1: factor 2/1

2. Similar-triangle comparison 2: factor 3/1

3. Similar-triangle comparison 3: factor 4/1

4. Similar-triangle comparison 4: factor 1/2

5. Similar-triangle comparison 5: factor 3/2

6. Similar-triangle comparison 6: factor 5/2

7. Similar-triangle comparison 7: factor 1/3

8. Similar-triangle comparison 8: factor 4/3

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 4/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

9. Similar-triangle comparison 9: factor 6/5

10. Similar-triangle comparison 10: factor 7/4

11. Similar-triangle comparison 11: factor 5/4

12. Similar-triangle comparison 12: factor 1/1

13. Similar-triangle comparison 13: factor 2/3

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 2/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

14. Similar-triangle comparison 14: factor 2/1

15. Similar-triangle comparison 15: factor 3/1

16. Similar-triangle comparison 16: factor 4/1

17. Similar-triangle comparison 17: factor 1/2

18. Similar-triangle comparison 18: factor 3/2

19. Similar-triangle comparison 19: factor 5/2

20. Similar-triangle comparison 20: factor 1/3

21. Similar-triangle comparison 21: factor 4/3

22. Similar-triangle comparison 22: factor 6/5

23. Similar-triangle comparison 23: factor 7/4

24. Similar-triangle comparison 24: factor 5/4

25. Similar-triangle comparison 25: factor 1/1

26. Similar-triangle comparison 26: factor 2/3

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 2/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

27. Similar-triangle comparison 27: factor 2/1

28. Similar-triangle comparison 28: factor 3/1

29. Similar-triangle comparison 29: factor 4/1

30. Similar-triangle comparison 30: factor 1/2

31. Similar-triangle comparison 31: factor 3/2

32. Similar-triangle comparison 32: factor 5/2

33. Similar-triangle comparison 33: factor 1/3

34. Similar-triangle comparison 34: factor 4/3

First possible misconception. A repeating third should be written as an exact fraction in a precisely stated similarity example. Writing a rounded four-decimal side length and then claiming that every visible ratio is exactly 4/3 is not logically exact. Also, similarity allows different sizes; congruence does not.

35. Similar-triangle comparison 35: factor 6/5

36. Similar-triangle comparison 36: factor 7/4

37. Similar-triangle comparison 37: factor 5/4

38. Missing corresponding side 1: derive the factor

Independent check. Reverse the conversion: 16/4=4 cm. The B perimeter would be 48 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.25, not 4. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

39. Missing corresponding side 2: derive the factor

Independent check. Reverse the conversion: 12.5/0.5=25 cm. The B perimeter would be 28 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 2, not 0.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

40. Missing corresponding side 3: derive the factor

Independent check. Reverse the conversion: 15/1.5=10 cm. The B perimeter would be 90 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.6667, not 1.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

41. Missing corresponding side 4: derive the factor

Independent check. Reverse the conversion: 50/2.5=20 cm. The B perimeter would be 150 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.4, not 2.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

42. Missing corresponding side 5: exact scale factor

Independent check. Reverse the result: (7/3)÷(1/3)=7 cm. The new perimeter would be 5 cm compared with original 15 cm, again preserving scale factor 1/3.

43. Missing corresponding side 6: exact scale factor

Independent check. Reverse the result: (4)÷(4/3)=3 cm. The new perimeter would be 16 cm compared with original 12 cm, again preserving scale factor 4/3.

44. Missing corresponding side 7: derive the factor

Independent check. Reverse the conversion: 28.8/1.2=24 cm. The B perimeter would be 67.2 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8333, not 1.2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

45. Missing corresponding side 8: derive the factor

Independent check. Reverse the conversion: 45.5/1.75=26 cm. The B perimeter would be 105 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5714, not 1.75. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

46. Missing corresponding side 9: derive the factor

Independent check. Reverse the conversion: 18.75/1.25=15 cm. The B perimeter would be 75 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8, not 1.25. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

47. Missing corresponding side 10: derive the factor

Independent check. Reverse the conversion: 5/1=5 cm. The B perimeter would be 15 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 1, not 1. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

48. Missing corresponding side 11: exact scale factor

Independent check. Reverse the result: (10/3)÷(2/3)=5 cm. The new perimeter would be 8 cm compared with original 12 cm, again preserving scale factor 2/3.

49. Missing corresponding side 12: derive the factor

Independent check. Reverse the conversion: 14/2=7 cm. The B perimeter would be 112 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5, not 2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

50. Missing corresponding side 13: derive the factor

Independent check. Reverse the conversion: 72/3=24 cm. The B perimeter would be 180 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.3333, not 3. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

51. Missing corresponding side 14: derive the factor

Independent check. Reverse the conversion: 100/4=25 cm. The B perimeter would be 240 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.25, not 4. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

52. Missing corresponding side 15: derive the factor

Independent check. Reverse the conversion: 1.5/0.5=3 cm. The B perimeter would be 7.5 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 2, not 0.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

53. Missing corresponding side 16: derive the factor

Independent check. Reverse the conversion: 6/1.5=4 cm. The B perimeter would be 18 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.6667, not 1.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

54. Missing corresponding side 17: derive the factor

Independent check. Reverse the conversion: 62.5/2.5=25 cm. The B perimeter would be 140 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.4, not 2.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

55. Missing corresponding side 18: exact scale factor

Independent check. Reverse the result: (10/3)÷(1/3)=10 cm. The new perimeter would be 20 cm compared with original 60 cm, again preserving scale factor 1/3.

56. Missing corresponding side 19: exact scale factor

Independent check. Reverse the result: (80/3)÷(4/3)=20 cm. The new perimeter would be 80 cm compared with original 60 cm, again preserving scale factor 4/3.

57. Missing corresponding side 20: derive the factor

Independent check. Reverse the conversion: 8.4/1.2=7 cm. The B perimeter would be 18 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8333, not 1.2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

58. Missing corresponding side 21: derive the factor

Independent check. Reverse the conversion: 5.25/1.75=3 cm. The B perimeter would be 21 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5714, not 1.75. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

59. Missing corresponding side 22: derive the factor

Independent check. Reverse the conversion: 30/1.25=24 cm. The B perimeter would be 70 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.8, not 1.25. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

60. Missing corresponding side 23: derive the factor

Independent check. Reverse the conversion: 26/1=26 cm. The B perimeter would be 60 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 1, not 1. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

61. Missing corresponding side 24: exact scale factor

Independent check. Reverse the result: (10)÷(2/3)=15 cm. The new perimeter would be 40 cm compared with original 60 cm, again preserving scale factor 2/3.

62. Missing corresponding side 25: derive the factor

Independent check. Reverse the conversion: 10/2=5 cm. The B perimeter would be 30 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.5, not 2. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

63. Missing corresponding side 26: derive the factor

Independent check. Reverse the conversion: 15/3=5 cm. The B perimeter would be 36 cm, compared with A’s 12 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.3333, not 3. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

64. Missing corresponding side 27: derive the factor

Independent check. Reverse the conversion: 28/4=7 cm. The B perimeter would be 224 cm, compared with A’s 56 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.25, not 4. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

65. Missing corresponding side 28: derive the factor

Independent check. Reverse the conversion: 12/0.5=24 cm. The B perimeter would be 30 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 2, not 0.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

66. Missing corresponding side 29: derive the factor

Independent check. Reverse the conversion: 37.5/1.5=25 cm. The B perimeter would be 90 cm, compared with A’s 60 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.6667, not 1.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

67. Missing corresponding side 30: derive the factor

Independent check. Reverse the conversion: 7.5/2.5=3 cm. The B perimeter would be 37.5 cm, compared with A’s 15 cm, preserving the same factor.

Transfer. Now imagine a copy from B back to A. The factor reverses to 0.4, not 2.5. Explain why moving between diagrams in opposite directions uses reciprocal scaling.

68. Missing corresponding side 31: exact scale factor

Independent check. Reverse the result: (4/3)÷(1/3)=4 cm. The new perimeter would be 4 cm compared with original 12 cm, again preserving scale factor 1/3.

69. Rectangle aspect-ratio check 1

Try this. Rectangle A measures 3 by 4 cm and rectangle B measures 6 by 8 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:4 and 6:8. They simplify to the same comparison; the area multiplier is 4.

70. Rectangle aspect-ratio check 2

Try this. Rectangle A measures 3 by 5 cm and rectangle B measures 9 by 15 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:5 and 9:15. They simplify to the same comparison; the area multiplier is 9.

71. Rectangle aspect-ratio check 3

Try this. Rectangle A measures 2 by 7 cm and rectangle B measures 4 by 14 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 2:7 and 4:14. They simplify to the same comparison; the area multiplier is 4.

72. Rectangle aspect-ratio check 4

Try this. Rectangle A measures 4 by 9 cm and rectangle B measures 8 by 18 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 4:9 and 8:18. They simplify to the same comparison; the area multiplier is 4.

73. Rectangle aspect-ratio check 5

Try this. Rectangle A measures 4 by 7 cm and rectangle B measures 6 by 10.5 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 4:7 and 6:10.5. They simplify to the same comparison; the area multiplier is 2.25.

74. Rectangle aspect-ratio check 6

Try this. Rectangle A measures 3 by 4 cm and rectangle B measures 6 by 9 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:4 and 6:9. They are different; a width factor 2 cannot simultaneously explain the height factor 2.25.

75. Rectangle aspect-ratio check 7

Try this. Rectangle A measures 2 by 5 cm and rectangle B measures 4 by 11 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 2:5 and 4:11. They are different; a width factor 2 cannot simultaneously explain the height factor 2.2.

76. Rectangle aspect-ratio check 8

Try this. Rectangle A measures 3 by 7 cm and rectangle B measures 6 by 12 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:7 and 6:12. They are different; a width factor 2 cannot simultaneously explain the height factor 1.7143.

77. Rectangle aspect-ratio check 9

Try this. Rectangle A measures 4 by 8 cm and rectangle B measures 10 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 4:8 and 10:20. They simplify to the same comparison; the area multiplier is 6.25.

78. Rectangle aspect-ratio check 10

Try this. Rectangle A measures 5 by 12 cm and rectangle B measures 10 by 25 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 5:12 and 10:25. They are different; a width factor 2 cannot simultaneously explain the height factor 2.0833.

79. Rectangle aspect-ratio check 11

Try this. Rectangle A measures 2 by 9 cm and rectangle B measures 4 by 15 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 2:9 and 4:15. They are different; a width factor 2 cannot simultaneously explain the height factor 1.6667.

80. Rectangle aspect-ratio check 12

Try this. Rectangle A measures 3 by 9 cm and rectangle B measures 6 by 18 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:9 and 6:18. They simplify to the same comparison; the area multiplier is 4.

81. Rectangle aspect-ratio check 13

Try this. Rectangle A measures 7 by 8 cm and rectangle B measures 14 by 16 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 7:8 and 14:16. They simplify to the same comparison; the area multiplier is 4.

82. Rectangle aspect-ratio check 14

Try this. Rectangle A measures 4 by 5 cm and rectangle B measures 8 by 10 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 4:5 and 8:10. They simplify to the same comparison; the area multiplier is 4.

83. Rectangle aspect-ratio check 15

Try this. Rectangle A measures 3 by 6 cm and rectangle B measures 10 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:6 and 10:20. They simplify to the same comparison; the area multiplier is 11.1111.

84. Rectangle aspect-ratio check 16

Try this. Rectangle A measures 3 by 8 cm and rectangle B measures 9 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:8 and 9:20. They are different; a width factor 3 cannot simultaneously explain the height factor 2.5.

85. Rectangle aspect-ratio check 17

Try this. Rectangle A measures 2 by 3 cm and rectangle B measures 6 by 9 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 2:3 and 6:9. They simplify to the same comparison; the area multiplier is 9.

86. Rectangle aspect-ratio check 18

Try this. Rectangle A measures 5 by 7 cm and rectangle B measures 10 by 13 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 5:7 and 10:13. They are different; a width factor 2 cannot simultaneously explain the height factor 1.8571.

87. Rectangle aspect-ratio check 19

Try this. Rectangle A measures 3 by 5 cm and rectangle B measures 12 by 20 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 3:5 and 12:20. They simplify to the same comparison; the area multiplier is 16.

88. Rectangle aspect-ratio check 20

Try this. Rectangle A measures 2 by 7 cm and rectangle B measures 6 by 18 cm in the matching orientation. Are they similar? What evidence would support or reject that claim?

Independent check. The shape ratios are 2:7 and 6:18. They are different; a width factor 3 cannot simultaneously explain the height factor 2.5714.

89. Congruent shapes need not face the same direction

Try this. A triangle is reflected in a mirror. Is its image congruent?

Worked reasoning. Yes. Reflection preserves all lengths and angles, so the image is congruent although its orientation reverses.

First possible misconception. Assuming rotated or reflected figures cannot be congruent because the drawing looks reversed.

Independent check. Trace or superimpose corresponding sides under the reflection.

90. Congruence implies similarity

Try this. If two figures are congruent, must they be similar?

Worked reasoning. Yes. Every corresponding side length ratio is one and angles match, so they are similar at scale factor one.

First possible misconception. Treating congruence and similarity as disjoint categories.

Independent check. Every congruent copy fits the similarity definition with k=1.

91. Similarity does not imply congruence

Try this. If two triangles have equal corresponding angles, must they be congruent?

Worked reasoning. No. They can be similar at different sizes, for example 3–4–5 and 6–8–10.

First possible misconception. Confusing equal angles with equal physical lengths.

Independent check. Compare all three side ratios.

92. Two squares are similar

Try this. Are two squares of different side lengths similar?

Worked reasoning. Yes. All corresponding angles are 90° and every side scales by the same length factor.

First possible misconception. Calling different-size squares congruent.

Independent check. Their lengths differ, but the shape remains a square.

93. Two rectangles need not be similar

Try this. Is every rectangle similar to every other rectangle?

Worked reasoning. No. A 2-by-3 rectangle has ratio 2:3, unlike a 2-by-5 rectangle.

First possible misconception. All right angles are not enough for polygon similarity.

Independent check. Compare corresponding side ratios.

94. Scale factor one preserves all lengths

Try this. A figure is enlarged by scale factor 1. Is its image congruent?

Worked reasoning. Yes, for an ordinary similarity transformation with factor one, all corresponding lengths and angles remain equal.

First possible misconception. Assuming the word enlargement always means a size increase.

Independent check. Factor one represents unchanged size, possibly moved or reflected.

95. A length scale factor affects perimeter once

Try this. A triangle perimeter is 24 cm and a similar copy has scale factor 3. Find new perimeter.

Worked reasoning. It is 3×24=72 cm. Every side is tripled.

First possible misconception. Squaring the factor for perimeter would be wrong.

Independent check. The three individual sides all triple, so their sum triples.

96. An area scales with the square of a length factor

Try this. A rectangle area is 12 cm² and all lengths are doubled. Find new area.

Worked reasoning. The length factor is two, area factor is four, so new area=48 cm².

First possible misconception. Doubling area to 24 cm² treats it like perimeter.

Independent check. A 3-by-4 rectangle becomes 6-by-8 with area 48.

97. Area ratio can recover a length factor

Try this. Two similar plane figures have areas in ratio 9:25. What is the corresponding length ratio?

Worked reasoning. For positive side scales, take square roots: length ratio is 3:5.

First possible misconception. Using 9:25 directly as the side ratio changes the area again.

Independent check. Squares of 3 and 5 reconstruct 9 and 25.

98. Congruent triangles have equal area

Try this. Two triangles are congruent, but one is rotated. Compare area.

Worked reasoning. Their areas match. Rigid motions preserve size and shape.

First possible misconception. Assuming the rotated triangle fills a different area.

Independent check. Superimpose after rotation.

99. Equal area does not prove congruence

Try this. A 3-by-4 rectangle and a 2-by-6 rectangle both have area twelve. Are they congruent?

Worked reasoning. No. They have different matching side lengths and different perimeters.

First possible misconception. Treating area equality as all-shape equality.

Independent check. Compute perimeters fourteen and sixteen.

100. Equal perimeter does not prove similarity

Try this. A 3-by-7 rectangle and a 4-by-6 rectangle both have perimeter twenty. Are they similar?

Worked reasoning. No. Their aspect ratios differ, despite equal perimeters.

First possible misconception. Using a single boundary total as proof of shape proportions.

Independent check. Compare 3:7 with 4:6.

101. Corresponding order matters

Try this. Triangle ABC corresponds to DEF in that order. Which sides match?

Worked reasoning. AB with DE, BC with EF and AC with DF.

First possible misconception. Matching sides based on their position on paper instead of named vertices.

Independent check. Use vertex A↔D, B↔E, C↔F.

102. Similar triangles have equal corresponding angles

Try this. Two triangles are similar. Angle A corresponds to D and angle A=45°. What is D?

Worked reasoning. Angle D is also 45°. Similarity preserves angles.

First possible misconception. Multiplying an angle by the length scale factor.

Independent check. Angles stay equal even when side lengths change.

103. Congruent triangles share every corresponding side length

Try this. Two triangles are congruent, with side AB=7 cm corresponding to DE. Find DE.

Worked reasoning. DE is 7 cm.

First possible misconception. Multiplying by an enlargement factor other than one contradicts congruence.

Independent check. The corresponding side ratio is exactly one.

104. A triangle’s shape is fixed by its three sides up to congruence

Try this. Two valid triangles have matching three side lengths 3,4,5 in correspondence. Are they congruent?

Worked reasoning. Yes, the side-side-side congruence property applies.

First possible misconception. Assuming mirror orientation changes side lengths.

Independent check. Rotate or reflect one to superimpose.

105. Triangles with equal angle sets are similar

Try this. One triangle has angles 40°,60°,80°; another has 60°,80°,40°. Are they similar?

Worked reasoning. Yes, the corresponding angle sets match and the triangles are similar.

First possible misconception. Matching angle labels in the displayed order without relabelling corresponding corners.

Independent check. Angle-angle similarity suffices for Euclidean triangles.

106. Angle equality alone is not enough for general quadrilaterals

Try this. Two rectangles have four right angles but different aspect ratios. Are they necessarily similar?

Worked reasoning. No. Side proportions must agree as well.

First possible misconception. Importing the triangle angle-angle shortcut into every polygon.

Independent check. Compare 2-by-3 and 2-by-5 rectangles.

107. A scale factor can be a fraction

Try this. A triangle side 12 cm becomes 9 cm in a similar smaller copy. Find scale factor.

Worked reasoning. k=9/12=3/4, less than one.

First possible misconception. Assuming an enlargement or similarity scale must exceed one.

Independent check. New side/old side=0.75.

108. Reverse a reduction

Try this. A reduced copy has side 6 cm at factor 1/2. Find original side.

Worked reasoning. Original=6/(1/2)=12 cm.

First possible misconception. Multiplying by 1/2 again reduces rather than reverses.

Independent check. 12×1/2=6.

109. Similar figures need compatible units

Try this. One matching side is 40 cm; another is 1.2 m. Find scale from first to second.

Worked reasoning. Convert 1.2 m to 120 cm. Factor=120/40=3.

First possible misconception. Comparing 40:1.2 before common units creates a false ratio.

Independent check. 40×3=120 cm=1.2 m.

110. A diagonal scales with every length

Try this. A 3-by-4 rectangle becomes 6-by-8. Compare diagonals.

Worked reasoning. Diagonals scale from 5 cm to 10 cm: factor two.

First possible misconception. Assuming diagonals can scale differently from sides in a similar figure.

Independent check. Pythagoras gives √(3²+4²)=5 and √(6²+8²)=10.

111. Changing only one side does not preserve shape

Try this. A 3-by-4 rectangle becomes 6-by-4. Are they similar?

Worked reasoning. No. Width doubled while height stayed constant, so the aspect ratio changed.

First possible misconception. Declaring similarity because one pair of sides has the right factor.

Independent check. Check 6/3=2 but 4/4=1.

112. One valid matched side pair does not prove similarity

Try this. Two arbitrary triangles have a 5-cm side each. Does that establish similarity?

Worked reasoning. No. Another two corresponding side ratios or angle information is needed.

First possible misconception. Using one matching feature as a complete proof.

Independent check. Two triangles could share one side yet have different other sides.

113. Scalability in maps

Try this. A fictional scale drawing uses 1 cm to 50 cm. A model shape side is 6 cm on drawing. What is its real length?

Worked reasoning. 6×50=300 cm, or 3 m.

First possible misconception. Treating 1:50 as 1 cm to 50 m changes the unit by a hundredfold.

Independent check. 300/50=6 drawing centimetres.

114. A scaled area on a plan

Try this. A flat shape drawn at 1:100 has drawing area 4 cm². What ideal actual area in m²?

Worked reasoning. Area factor is 100²=10,000; real area 40,000 cm²=4 m².

First possible misconception. Multiplying by only 100 for area ignores two dimensions.

Independent check. A 2-by-2 cm square corresponds to 2-by-2 m, area 4 m².

115. Area ratio and perimeter ratio differ

Try this. Similar figures have length scale 3/2. State perimeter and area scale factors.

Worked reasoning. Perimeter factor 3/2; area factor (3/2)²=9/4.

First possible misconception. Applying 3/2 to area without squaring.

Independent check. Compare a square of side two with side three: areas four and nine.

116. Similarity preserves angles but not necessarily area

Try this. A triangle is enlarged by scale factor four. What happens to one 70° angle and its area?

Worked reasoning. Angle stays 70°; area becomes sixteen times its original.

First possible misconception. Multiplying angle by four would give an unsupported 280° corner.

Independent check. Similarity is an angle-preserving transformation with length scaling.

117. Some similar figures are congruent

Try this. Can similar figures also be congruent?

Worked reasoning. Yes, if their positive scale factor is one.

First possible misconception. Answering ‘never’ confuses broader and narrower shape categories.

Independent check. Identical-sized copies satisfy both definitions.

118. Congruence does not require the same drawing position

Try this. Two identical triangles are translated horizontally. Are they still congruent?

Worked reasoning. Yes. Translation changes position but no length or angle.

First possible misconception. Demanding identical coordinates for matching figures.

Independent check. Rigid motion gives an exact overlap after moving one.

119. Area factor works for triangles too

Try this. A triangle has area 10 cm² and a similar copy has length factor 2.5. Find new area.

Worked reasoning. Area factor 2.5²=6.25. New area=62.5 cm².

First possible misconception. Using a linear factor of 2.5 for area gives 25, too small.

Independent check. Compare base and perpendicular height, each multiplied by 2.5.

120. The shortest side should match the shortest side

Try this. Two similar triangles have side lists 4,6,9 and 8,12,18. Which sides correspond naturally?

Worked reasoning. 4↔8, 6↔12, 9↔18; all ratios two.

First possible misconception. Matching four with eighteen creates inconsistent ratios.

Independent check. Sort sides or use supplied vertex correspondence.

121. An image resized nonuniformly is distorted

Try this. A rectangle image is stretched twice as wide but unchanged in height. Is the new image similar?

Worked reasoning. No, except in degenerate cases. The aspect ratio changes, so it is a distortion rather than a uniform similarity.

First possible misconception. Calling any bigger picture a proportional enlargement.

Independent check. Compare horizontal and vertical scaling separately.

122. Angles of similar quadrilaterals

Try this. Two quadrilaterals are given similar, and a corresponding angle in one is 105°. Find the second.

Worked reasoning. It is 105°, because similarity preserves corresponding angle size.

First possible misconception. Scaling angles using a side factor.

Independent check. Identify corresponding vertices and apply equality.

123. Congruent isn’t the same as equal perimeters alone

Try this. Two different rectangles both have 20-cm perimeter. Must they be congruent?

Worked reasoning. No. 3-by-7 and 4-by-6 both have perimeter twenty but different side lengths.

First possible misconception. Boundary equality is not an entire shape description.

Independent check. A congruence claim needs matched geometry.

124. Area formula can expose a mismatch

Try this. A right triangle with legs three and four is compared to one with legs six and eight. What is their area ratio?

Worked reasoning. Areas are six and twenty-four square units, so ratio 1:4; length ratio 1:2.

First possible misconception. Using a length ratio 1:2 directly as area ratio.

Independent check. Both perpendicular legs double, giving four times the area.

125. A scale of one-third reverses with three

Try this. A side 15 cm becomes 5 cm. State downscale and reverse factors.

Worked reasoning. From original to copy k=1/3; from copy to original factor three.

First possible misconception. Using 1/3 in both directions.

Independent check. 5×3=15 and 15×1/3=5.

Four learning gaps with different repairs

The learner assumes similar means same size

The wrong side pairs are compared

Area is scaled by the length factor instead of its square

A single shared angle is treated as complete proof

How a three-pupil G1 Mathematics lesson can work

10 minutes: retrieve equal angles, parallel marks and common ratios.
20 minutes: compare a congruent and a scaled triangle with labelled correspondence.
25 minutes: calculate missing sides and check matching ratios independently.
15 minutes: compare perimeter versus area scaling and rectangular aspect ratios.
10 minutes: solve an unseen side correspondence or counterexample question.
10 minutes: explain the first mismatch and set a later retest.

Five short practice sessions

Day 1: identify congruent copies under translation, rotation and reflection.
Day 2: match corresponding sides in two named similar triangles.
Day 3: find new side lengths under whole-number and fractional scale factors.
Day 4: compare perimeter and area factors without memorising the wrong one.
Day 5: find a rectangle counterexample that shows identical angles alone are not sufficient for similarity.

Subject-level and examination-year accuracy

Frequently asked questions

Are congruent shapes also similar?

Yes. Their similarity scale factor is one and all matching measurements agree.

Are all similar shapes congruent?

No. Similar shapes may differ in size; only those with scale factor one are congruent.

Does rotating a shape stop it being congruent?

No. Rotation preserves length and angle. So do translation and reflection.

Why is area factor k²?

Area has two dimensions, both scaling by k. The combined factor is k×k=k².

Are all rectangles similar?

No. Their corresponding side ratios must agree; rectangles can have different aspect ratios.

What should a tutor check beyond the final side answer?

The matched vertex order, scale direction, uniform side ratios and whether the conclusion is similarity or congruence.

Official curriculum and reader notes

Bring one incorrect matching-side diagram to eduKate Singapore and ask how a tutor would teach corresponding sides and scale factors before expanding to harder geometry.

Secondary 2 G1 Mathematics Tutor Bukit Timah | Congruence, Similarity and Scale Factors

Official curriculum and reader notes

Official curriculum and reader notes

Subject-level and examination-year accuracy

Frequently asked questions

Are congruent shapes also similar?

Are all similar shapes congruent?

Does rotating a shape stop it being congruent?

Why is area factor k²?

Are all rectangles similar?

What should a tutor check beyond the final side answer?

Continue relevant Bukit Timah G1 Maths guides

Official curriculum and reader notes

Subject-level and examination-year accuracy

Frequently asked questions

Are congruent shapes also similar?

Are all similar shapes congruent?

Does rotating a shape stop it being congruent?

Why is area factor k²?

Are all rectangles similar?

What should a tutor check beyond the final side answer?

Continue relevant Bukit Timah G1 Maths guides

Official curriculum and reader notes

Five short practice sessions

Subject-level and examination-year accuracy

Frequently asked questions

Are congruent shapes also similar?

Are all similar shapes congruent?

Does rotating a shape stop it being congruent?

Why is area factor k²?

Are all rectangles similar?

What should a tutor check beyond the final side answer?

Continue relevant Bukit Timah G1 Maths guides

Official curriculum and reader notes

Five short practice sessions

Subject-level and examination-year accuracy

Frequently asked questions

Are congruent shapes also similar?

Are all similar shapes congruent?

Does rotating a shape stop it being congruent?

Why is area factor k²?

Are all rectangles similar?

What should a tutor check beyond the final side answer?

Continue relevant Bukit Timah G1 Maths guides

Official curriculum and reader notes

The one-minute comparison

Similarity preserves shape, congruence preserves size too

Congruence tolerates movement and reflection

Similarity requires one multiplier for matching lengths

Match corresponding vertices before dividing

Why an area does not scale like a side length

125 original correspondence and scale-factor clinics

1. Similar-triangle comparison 1: factor 2/1

2. Similar-triangle comparison 2: factor 3/1

3. Similar-triangle comparison 3: factor 4/1

4. Similar-triangle comparison 4: factor 1/2

5. Similar-triangle comparison 5: factor 3/2

6. Similar-triangle comparison 6: factor 5/2

7. Similar-triangle comparison 7: factor 1/3

8. Similar-triangle comparison 8: factor 4/3

9. Similar-triangle comparison 9: factor 6/5

10. Similar-triangle comparison 10: factor 7/4

11. Similar-triangle comparison 11: factor 5/4

12. Similar-triangle comparison 12: factor 1/1

13. Similar-triangle comparison 13: factor 2/3

14. Similar-triangle comparison 14: factor 2/1

15. Similar-triangle comparison 15: factor 3/1

16. Similar-triangle comparison 16: factor 4/1

17. Similar-triangle comparison 17: factor 1/2

18. Similar-triangle comparison 18: factor 3/2

19. Similar-triangle comparison 19: factor 5/2

20. Similar-triangle comparison 20: factor 1/3

21. Similar-triangle comparison 21: factor 4/3

22. Similar-triangle comparison 22: factor 6/5

23. Similar-triangle comparison 23: factor 7/4

24. Similar-triangle comparison 24: factor 5/4

25. Similar-triangle comparison 25: factor 1/1

26. Similar-triangle comparison 26: factor 2/3

27. Similar-triangle comparison 27: factor 2/1

28. Similar-triangle comparison 28: factor 3/1