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Pythagoras’ theorem practice questions

Pythagoras' theorem states that in a right-angled triangle, a squared plus b squared equals c squared, where c is the hypotenuse. To find the hypotenuse, square both shorter sides, add them, then take the square root. For legs of 5cm and 12cm, the hypotenuse is 13cm.

Pythagoras’ theorem connects the three sides of any right-angled triangle: a² + b² = c², where c is the hypotenuse, the side opposite the right angle. It appears on every GCSE paper and underpins trigonometry, so it is worth knowing cold.

Year 9GCSE FoundationGCSE Higher

The method, step by step

  1. Identify the hypotenuse. It is always the longest side, and always opposite the right angle.
  2. Decide what you are finding. Looking for the hypotenuse means adding the squares. Looking for a shorter side means subtracting.
  3. Square the known sides. Do this before anything else.
  4. Add or subtract. Add if finding the hypotenuse, subtract if finding a shorter side.
  5. Square root. The final step, easy to forget under exam pressure.

Worked example

A right-angled triangle has legs of 5cm and 12cm. Find the hypotenuse.
Square both known sides: 5² = 25 and 12² = 144.
The hypotenuse is being found, so add: 25 + 144 = 169.
Square root the total: √169 = 13.
The hypotenuse is 13cm

Common mistakes

Adding when you should subtract. If the hypotenuse is already known, you subtract to find the missing shorter side.

Stopping at the squared value. The answer 169 is c², not c. Always take the square root.

Assuming the hypotenuse is the side drawn longest on the page. Diagrams are rarely to scale, so find the right angle instead.

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