Remember Pythagoras' Theorem?

Pythagoras' Theorem

$a^2+b^2=c^2$`a`2+`b`2=`c`2, where

- $c$
`c` is the length of the hypotenuse, and
- $a$
`a` and $b$`b` are the lengths of the other two sides

We can rearrange this equation to find formulas for each side length.

Rearranging Pythagoras' Theorem

To find the hypotenuse: $c=\sqrt{a^2+b^2}$`c`=√`a`2+`b`2

To find a shorter side: $a=\sqrt{c^2-b^2}$`a`=√`c`2−`b`2

To apply Pythagoras' Theorem to real life situations,

- Look for right-angled triangles
- Choose which side, hypotenuse or a shorter side, you are trying to find
- Find the lengths of the other two sides
- Apply the relevant formula and substitute the lengths of the other two sides

Let's look at some examples so we can see this in action.

#### Worked Examples

##### Question 1

Consider a cone with slant height $13$13m and perpendicular height $12$12m.

Find the length of the radius, $r$`r`, of the base of this cone.

Hence, find the length of the diameter of the cone's base.

##### Question 2

Find the length of the unknown side, x, in the given trapezium.

Give your answer correct to $2$2 decimal places.