# Finding a Shorter Side Using Pythagoras' Theorem

(KS3, Year 7)

In the formula,

**a**and

**b**are the lengths of the shorter sides and

**c**is the length of the hypotenuse. The image below shows what we mean:

## How to Find the Hypotenuse Using Pythagoras' Theorem

## Question

What is the unknown length, a, of the right triangle below?## Step-by-Step:

## 1

Start with the formula:

a = √(c

^{2}− b^{2})**Don't forget:**√ means square root**and**c^{2}= c × c (c squared), b^{2}= b × b.## 2

Find the lengths of the sides from the right triangle.
In our example, the known side lengths are b = 6 and h = 10.

## 3

Substitute b = 6 and c = 10 into the formula.

a = √(10^{2} − 6^{2}) = √((10 × 10) − (6 × 6))

c = √(100 − 36) = √64

a = 8

## Answer:

The length of the shorter side, a, is 8.## A Real Example of How to Find a Shorter Side Using Pythagoras' Theorem

In the example above, we have used the formula**a = √(c**to find the length of the side, a. This is just a rearrangement of the more memorable formula,

^{2}− b^{2})**a**(see

^{2}+ b^{2}= c^{2}**Note**). If you find this simpler formula easier to remember, use it! Substitute in the lengths you know (replace the letters b and c with numbers) and then rearrange to find the side, a. There are two shorter sides, a and b. It does not really matter which of the shorter sides is labelled a and which is labelled b. The formula

**b = √(c**can also be used.

^{2}− a^{2})## Interactive Widget

Here is an interactive widget to help you learn about Pythagoras' theorem.## Rearranging Pythagoras' Theorem

Pythagoras' theorem states:Subtract

**b**from both sides:

^{2}Take square roots of both sides:

This gives the formula for finding the length of the unknown shorter side when the hypotenuse and the other shorter side are known:

## Top Tip

## Leaving the Answer as a Square Root

To find the side, a, you have to find the square root of a^{2}. Unless a

^{2}is a square number, a will not be a whole number. It is sometimes best to leave the answer as a square root (also called a surd). For example, find the hypotenuse for the right triangle below, where b = 1 and c = 2:

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

a^{2} + 1^{2} = 2^{2}

a^{2} = 2^{2} − 1^{2}

a^{2} = 4 − 1

a^{2} = 3

## Worksheet

This test is printable and sendable