Calculating the length of another side of a triangle

If you know the length of the hypotenuse and one of the other sides, you can use Pythagoras’ theorem to find the length of the third side.

We can rearrange the formula for Pythagoras’ theorem, in order to make {b} or {c} the subject of the formula:

Diagram of a right-angled triangle

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

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

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

Example

Work out the length of the line {LM}, correct to {1} decimal place.

Diagram of a right-angled triangle with the values 4cm and 6cm

{LM}^{2}={LN}^{2}-{MN}^{2}

{LM}^{2}={6}^{2}-{4}^{2}

{LM}^{2}={36}-{16}

{LM}^{2}={20}

{LM}=\sqrt{20}

{LM}={4.5~cm} ( {1} decimal place)

Question

Work out the length of the line YZ, correct to {1} decimal place.

Right-angled triangle XYZ

Method 1

{YZ}^{2}+{XZ}^{2}={XY}^{2}

{YZ}^{2}+{7}^{2}={8}^{2}

{YZ}^{2}+{49}={64}

{YZ}^{2}={15} (subtract {49} from both sides)

{YZ}=\sqrt{15} (find the square root of both sides)

{YZ}={3.9~cm} ( {1} decimal place)

Method 2

{YZ}^{2}={XY}^{2}-{XZ}^{2}

{YZ}^{2}={8}^{2}-{7}^{2}

{YZ}^{2}={64}-{49}

{YZ}^{2}={15}

{YZ}=\sqrt{15} (find the square root of both sides)

{YZ}={3.9~cm} ( {1} decimal place)