Does your child use Pythagoras’ theorem whenever a question shows a triangle? The theorem applies to right-angled triangles. Before writing a² + b² = c², the learner must identify a 90° angle and the hypotenuse opposite it.
The error is usually a recognition problem, not a squaring problem. A familiar formula is being selected before the diagram’s conditions are checked. The repair is a short gate: right angle first, hypotenuse second, equation third.
Try this immediately: show a 3–4–5 right triangle beside an equilateral triangle with sides 3, 3 and 3. Ask your child to mark the right angle, name the hypotenuse and test the squares. The first gives 3² + 4² = 5²; the second does not satisfy 3² + 3² = 3².
Choose the triangle help you need
SECTION 1 OF 5
Pythagoras is a right-triangle theorem
For a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. If the legs are a and b and the hypotenuse is c, then a² + b² = c².
| Question | Decision |
|---|---|
| Is there a 90° angle? | If no, do not apply the ordinary Pythagorean theorem directly |
| Which side is opposite 90°? | That side is the hypotenuse, c |
| Are a and b the two legs? | Then a² + b² = c² |
| Are three side lengths given? | Use the longest as c when checking the converse |
A triangle can look almost right-angled without being one. Do not infer 90° from the sketch alone unless the right angle is marked, stated or can be established from given information.
The side letters are not fixed by position on the page. The hypotenuse can slope left, right or appear horizontal. Its defining feature is that it lies opposite the 90° angle and is the longest side of a non-degenerate right triangle.
SECTION 2 OF 5
Name the sides before substituting numbers
Many mistakes occur because the learner assumes c is whichever side is unknown. That is false. c represents the hypotenuse in the standard formula, whether it is known or unknown.
- If the unknown is the hypotenuse: c = √(a² + b²).
- If the unknown is a leg: a = √(c² − b²), or use the corresponding leg label.
- If no right angle is established: pause and choose another method.
Original teaching illustration
A right triangle has hypotenuse 13 cm and one leg 5 cm. The other leg is not √(13² + 5²). Since 13 is the hypotenuse, write 5² + b² = 13². Then b² = 169 − 25 = 144, so b = 12 cm.
A length is positive, so the contextual answer is 12 cm. The negative square root may appear when solving x² = 144 algebraically, but it is not a physical side length in this triangle.
SECTION 3 OF 5
The converse checks whether a triangle is right-angled
The theorem finds a missing side in a known right triangle. Its converse can test a triangle when all three side lengths are given: if the square of the longest side equals the sum of the squares of the other two, the triangle is right-angled.
- 5, 12, 13: 5² + 12² = 25 + 144 = 169 = 13², so it is right-angled.
- 4, 5, 6: 4² + 5² = 41, while 6² = 36, so this test does not identify a right triangle.
- 3, 3, 3: 3² + 3² ≠ 3², so the equilateral triangle is not right-angled.
Always choose the longest side as the candidate hypotenuse in a converse check. Testing a shorter side as c can produce a meaningless comparison.
At more advanced levels, non-right triangles may be solved with other geometry or trigonometry tools. The absence of a right angle is not a dead end; it is a signal to select a method whose conditions actually fit.
SECTION 4 OF 5
Replace formula spotting with a condition check
A diagnosis-first lesson asks why the formula was chosen. Listen for shortcuts such as “there are three sides”, “there is a square root” or “the question asks for length”. None of those conditions is sufficient.
- Mark: Find or prove the 90° angle.
- Name: Label the hypotenuse and the two legs.
- Model: Write the equation before inserting values.
- Solve: Rearrange carefully and calculate.
- Check: The hypotenuse should remain the longest side.
Use mixed diagrams with rotated right triangles and non-right triangles. The learner should sometimes decide not to use Pythagoras. A practice set that contains only valid right triangles cannot test method selection.
If a correct answer cannot be explained, ask the child to point to the 90° condition and name each side. Formula recall without condition recognition is fragile.
SECTION 5 OF 5
What useful Maths tuition should change
Progress appears when the student checks the diagram before reaching for the formula. The working should show the correct hypotenuse, a valid equation, accurate arithmetic and a reasonable length.
- Can the learner state the theorem’s right-angle condition?
- Can the learner find the hypotenuse in a rotated diagram?
- Can the learner rearrange for an unknown leg?
- Can the learner use the converse with the longest side?
- Can the learner reject Pythagoras when the triangle is not known to be right-angled?
In a 3-pax lesson, each learner can justify a method choice on a different triangle. The tutor can hear whether the gap is geometry vocabulary, formula selection, algebraic rearrangement or square-root calculation.
Match examples to the student’s actual Mathematics subject level and applicable syllabus. Posting Groups and subject levels are different, and not every subject is offered at every G level. For SEC candidates from 2027, verify the subject, level, syllabus year and candidate type.
Bring the triangle where the formula was chosen
One original diagram and the child’s first line of working are enough to reveal whether the issue is the right-angle condition, the hypotenuse, rearrangement or calculation.
eduKate’s verified positioning is 3-pax tuition. Please confirm suitable arrangements directly; no location, timetable, price, programme or result is promised here.
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Sources and scope
The decision table, examples and lesson routine are original teaching illustrations. The right-triangle condition and converse were checked against OpenStax. Policy and examination statements were checked against current MOE and SEAB information on 11 October 2026.