## Key points

- Pythagoras’ theorem states that for any right-angled triangle, the area of the square on the is equal to the sum of the areas of the squares on the other two sides.
- It can be thought of as \(a\)² + \(b\)² = \(c\)² where \(a\) and \(b\) are the shorter sides of the triangle, and \(c\) is the hypotenuse (longest side).
- Pythagoras’ theorem is only true for right-angled triangles. It is possible to check if a triangle is right-angled by in the lengths of the sides and seeing if the value of \(a\)² + \(b\)² is the same as the value of \(c\)².
- Pythagoras’ theorem can be used to find a missing side of a right-angled triangle. To find the hypotenuse, the values of \(a\)² and \(b\)² into the equation, and solve to find \(c\). This will involve adding the two squares and finding the of the answer.
- To find a shorter side, substitute the values into the equation and solve to find \(a\) or \(b\). This will involve subtracting the two squares and finding the square root of the answer.
- An understanding of
**powers and roots**is essential before exploring this topic.

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## What is Pythagoras' theorem?

Pythagoras’ theorem is a statement that is true for all right-angled triangles.It states that the area of the square on the is equal to the sum of the area of the squares on the other two sides.

It is useful to think of Pythagoras’ theorem as \(a\)² + \(b\)² = \(c\)².

The hypotenuse is labelled as \(c\) and the other two sides labelled as \(a\) and \(b\). This makes the areas of the squares \(c\)², \(a\)² and \(b\)².

### Examples

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### Question

Show that the triangle is a right-angled triangle using Pythagoras’ theorem.

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## Finding the length of the hypotenuse

can be used to calculate a missing side in a right-angled triangle. Follow these steps to find the length of the when the other two sides are given.

Label the sides \(a\), \(b\) and \(c\). Remember, the hypotenuse should always be labelled \(c\).

Then the values of \(a\) and \(b\) into the equation \(a\)² + \(b\)² = \(c\)².

Add the squares together to get the value of \(c\)².

Square root the value of \(c\)² to get the value of \(c\).

### Examples

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### Question

Find the length ST to 1 decimal place.

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## Finding the length of another side

Follow these steps to find the length of a side that is not the hypotenuse.

Label the sides \(a\), \(b\), and \(c\). Remember the should always be labelled \(c\)

Then the values of \(a\) and \(b\) into the equation \(a\)² + \(b\)² = \(c\)²

If finding \(b\), subtract the squares to get the value of \(b\)²

Finally, the value of \(b\)² to get the value of \(b\)

### Examples

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### Question

Find the length **EF**.

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## Practise using Pythagoras' theorem

### Quiz

Practise calculating different lengths of sides using Pythagoras' theorem with this quiz. You may need a pen and paper to help you with your answers.

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## Real-life maths

Flat screen televisions are usually measured diagonally from opposite corners of the screen.

This means that a 55 inch television does not have a width of 55 inches.It would measure 55 inches from the top left corner to the bottom right corner (as opposite.)

Pythagoras’ theorem can be used to calculate the diagonal size of the television screen, if the width and the height are known.

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### Game - Divided Islands

Divided Islands. gameDivided Islands

Use your maths skills to help the islanders of Ichi build bridges and bring light back to the islands in this free game from BBC Bitesize.

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