# Pythagoras' theorem · Year 9

> Pythagoras' theorem, Year 9: Pythagoras' theorem in shapes and more. Explanation, worked examples and free practice for the national curriculum in England.

In Pythagoras' theorem (Year 9) you practise 3 skills: Finding the hypotenuse and a shorter side · Pythagoras' theorem in shapes · Is it a right triangle?.

In a right triangle, the side opposite the right angle is called the hypotenuse. Pythagoras' theorem says: the squares of the two shorter sides add up to the square of the hypotenuse.

The theorem helps you find lengths in shapes: the diagonal of a rectangle is the hypotenuse of a triangle whose shorter sides are the sides of the rectangle. The height of an isosceles triangle makes a right triangle too.

Sometimes the hypotenuse is not a whole number. For example, if the shorter sides are 5 and 6, the hypotenuse squared is 61. The square root of 61 is between 7 and 8, and it is about 7.8.

The theorem also works the other way round: if the squares of the two shorter sides add up to the square of the longest side, the triangle has a right angle. That is how you can make a right angle with a rope: 3 metres, 4 metres and 5 metres.

**Tip:** The hypotenuse is always the longest side. If your hypotenuse comes out shorter than one of the other sides, check your working.

## What you learn

### Finding the hypotenuse and a shorter side

- Example: In a right triangle, the two shorter sides are 5 and 12. How long is the hypotenuse? (A right triangle with its two shorter sides, and the hypotenuse marked with a question mark.)
- Answer: 13
- Solution: 5² + 12² = 25 + 144 = 169 The hypotenuse squared is 169, so the hypotenuse is the square root of 169: √169 = 13

### Pythagoras' theorem in shapes

- Example: The sides of a rectangle are 12 cm and 9 cm. How long is its diagonal? (A rectangle with a dashed diagonal between two opposite corners. The length of the diagonal is marked with a question mark.)
- Answer: 15 cm
- Solution: By Pythagoras' theorem: 12² + 9² = 144 + 81 = 225 The diagonal squared is 225, so: √225 = 15 The diagonal is 15 cm long.

### Is it a right triangle?

- Example: The sides of a triangle are 6, 12 and 8. Does the triangle have a right angle?
- Options: Yes · No
- Answer: No
- Solution: Check: the sum of the squares of the two shorter sides, against the square of the longest side: 6² + 8² = 36 + 64 = 100 12² = 144 100 is not equal to 144, so the triangle has no right angle.

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

[Practise on the site](https://kixplo.com/uk/en/) · [All Year 9 topics](https://kixplo.com/uk/en/maths/year-9/)