To learn Pythagoras’ theorem quickly and correctly, begin by identifying the right angle in a triangle. The theorem compares the squares of the three side lengths of a right-angled triangle: the square on the longest side equals the sum of the squares on the two shorter sides. Learn this relationship with a picture, then use it to find missing lengths and check answers. These eduKateSG Mathematics Quick Notes explain when to use the theorem, how to avoid mixing up the sides and how to solve everyday problems clearly.
- Find the right angle: the theorem applies to right-angled triangles.
- Find the hypotenuse: it is the side opposite the right angle, and it is the longest.
- Label: write a² + b² = c², with c as the hypotenuse.
- Choose the right operation: add squares to find c; subtract squares to find a shorter side.
- Take the non-negative square root: a length cannot be negative.
- Check: the longest side and the squares must satisfy the theorem.
First, recognise the special triangle. A right-angled triangle has one angle of 90°. The side directly opposite that angle is the hypotenuse. You may draw a large or small right-angled triangle, rotate it or place it on its side; the hypotenuse is still opposite the 90° angle. Call the two shorter sides a and b, and the hypotenuse c. Then a² + b² = c². This formula describes a relationship among the squares of side lengths, not a + b = c.
Second, learn the simplest complete example. Suppose the shorter sides measure 3 cm and 4 cm. Square each: 3² = 9 and 4² = 16. Add them: 9 + 16 = 25. Since c² = 25 and c is a length, c = √25 = 5 cm. This is the well-known 3–4–5 right triangle. Check: 3² + 4² = 5², or 9 + 16 = 25. The arithmetic is short because you started with the correct side labels.
Third, understand why finding a shorter side uses subtraction. If the hypotenuse is 13 cm and one shorter side is 5 cm, let the missing side be x. Then x² + 5² = 13², so x² + 25 = 169. Subtract 25: x² = 144. Take the non-negative square root: x = 12 cm. Check that 5² + 12² = 13². When the question gives you the longest side, do not add its square to another square and make an even longer missing side.
Worked example: a ladder against a wall
A straight ladder is 10 metres long. Its base is 6 metres from a vertical wall, and the ground is level. How high up the wall does the ladder reach? The ladder is the hypotenuse because it lies opposite the right angle between wall and ground. Let the height be h: h² + 6² = 10². So h² = 100 − 36 = 64, giving h = 8 metres. Check that 6² + 8² = 10². State the assumptions if they are not given: perpendicular wall, level ground and a straight ladder.
Worked example: diagonal distance on coordinate axes
A point is at (1, 2), and another is at (4, 6). How far apart are they? Horizontal change = 4 − 1 = 3 units. Vertical change = 6 − 2 = 4 units. These perpendicular changes are the shorter sides of a right-angled triangle. The direct distance is √(3² + 4²) = √25 = 5 units. This is one reason Pythagoras’ theorem is useful in coordinate geometry, map grids and distance calculations.
The formula, read carefully
- c² = a² + b²: use when finding the hypotenuse.
- a² = c² − b²: use when finding a shorter side.
- c = √(a² + b²): give a positive length.
- a = √(c² − b²): use the positive square root for a physical length.
- Converse: if three positive side lengths satisfy a² + b² = c², with c the largest, the triangle is right-angled.
When the theorem does not apply
Do not apply the right-triangle formula to a triangle unless you know it is right-angled or are using the converse to test whether it is. For a triangle with sides 5 cm, 6 cm and 7 cm, the squares of the two shorter sides sum to 25 + 36 = 61, whereas the square of the longest side is 49. The two are not equal, so this is not a right-angled triangle. The formula is a relationship with a condition, not a general rule for every triangle.
The correct sequence for an exam question
- Draw or mark the triangle; locate the 90° angle.
- Circle the hypotenuse, the side opposite 90°.
- Write the formula with the hypotenuse on its own side of the equals sign.
- Substitute the known lengths and square them.
- Add or subtract to isolate the unknown squared length.
- Take the positive square root, attach units and verify the result.
Common mistakes and repairs
- Assuming the longest side is one of the short sides: identify the side opposite 90° first.
- Using a² + b² = c: remember that the right side is c², not c.
- Adding when finding a shorter side: subtract from the square of the hypotenuse.
- Taking √(a² + b²) as a + b: square first, add, then find the square root.
- Using the method without a right angle: state or establish the necessary condition.
- Forgetting units: the final answer is a length in cm, m or another appropriate unit, not a square unit.
Mini-test: explain the first step before calculating
- A right triangle has perpendicular sides 8 cm and 15 cm. Find its hypotenuse.
- A right triangle has hypotenuse 17 cm and one shorter side 8 cm. Find the other side.
- Are sides 6 cm, 8 cm and 10 cm consistent with a right triangle? Explain.
Answers and reasons
1. c² = 8² + 15² = 64 + 225 = 289, so c = 17 cm. 2. x² = 17² − 8² = 289 − 64 = 225, so x = 15 cm. 3. Yes: 6² + 8² = 36 + 64 = 100 = 10². The longest side is 10 cm, so the converse confirms a right triangle.
An eight-minute way to remember the theorem
Draw one right triangle, label the right angle and hypotenuse, then solve a 3–4–5 example. Change the numbers to a 5–12–13 example and find a shorter side. Use the last two minutes to check both results by squaring. Return tomorrow and solve a similar question with the triangle rotated. A correct method should survive a change of drawing, not depend on recognising one familiar picture.
Where to go next
Continue with Square Roots and Cube Roots · Perimeter and Area · Coordinate Graphs · Angles. Each Quick Notes article teaches one small topic. Read, try a question without looking, check and repeat tomorrow.
For a longer guide to mathematical foundations, see the immutable reference Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. The original page has not been changed.