# Geometry

To calculate the length of a side on a right-angled when you know the sizes of the other two, you need to use Pythagoras' Theorem.

Pythagoras' Theorem says that, in a right angled triangle:

The square of the is equal to the sum of the squares on the other two sides.

We can write this more simply as :

${a^2} = {b^2} + {c^2}$

## Calculating the length of the hypotenuse

Question

Use Pythagoras' Theorem to calculate the length of the hypotenuse. Give your answer to 2 decimal places.

Write the equation $$x^{2} = 7^{2} + 4^{2}$$

Square the lengths you know$$x^{2} = 49 + 16$$

Add together$$x^{2} = 65$$

Find the square root$$x = \sqrt {65}$$

$x = 8.06 (to\,2\,d.p.)$

Question

Calculate the length of side $$x$$

$x^{2} = 5^{2} + 9^{2}$

$x^{2} = 25 + 81$

$x^{2} = 106$

$x = \sqrt {106}$

$x = 10.30 (to\, 2\,d.p.)$

### Example

Calculate the length of the side marked $$a$$.

• Write the equation: $${12^2} = {a^2} + {8^2}$$
• Organise the equation $${a^2} = {12^2} - {8^2}$$. To find the length of a short side, we can also use the formula $${b^2} = {a^2} - {c^2}$$
• Square the lengths you know: $${a^2} = 144 - 64$$
• Do the subtraction: $${a^2} = 80$$
• Find the square root: $$a = \sqrt {80}$$
• $a = 8.94\,(to\,2\,d.p.)$

Question

Calculate the length of side a.

$a^{2} = 13^{2} - 9^{2}$
$a^{2} = 169 - 81$
$a^{2} = 88$
$a = \sqrt 88$
$a = 9.38$