Pythagoras’ theorem allows us to calculate lengths in right-angled triangles. Right-angled triangles are seen in everyday life – from the dimensions of a television to a ladder resting against a wall.

Pythagoras’ theorem allows us to calculate the length of any side of a right-angled triangle given the other two.

The square of the hypotenuse is equal to the sum of the squares of the remaining two sides.Pythagoras’ theorem

The hypotenuse is the longest side – it will always be opposite the right angle.

To represent this in a mathematical formula we can say;

\[{a}{^2}~=~{b}{^2}~{+}~{c}{^2}\]

Where \(a\) is the length of the hypotenuse and the other sides are labelled \(b\) and \(c\).

In this triangle we need to find the hypotenuse.

Pythagoras’ theorem tells us that:

\[{x}{^2}~=~{3}{^2}~{+}~{4}{^2}\]

\[{x}{^2}~=~{9}~{+}~{16}\]

\[{x}{^2}~=~{25}\]

To find \({x}\), we need to square root both sides added together.

\({x}\) = \(\sqrt{25}\)

\[{x}~{=}~{5}~{cm}\]

- Question
Find the length AC, giving your answer to two decimal places.

AC

^{2}= 6^{2}+ 2^{2}AC

^{2}= 36 + 4AC

^{2}= 40AC = \(\sqrt{40}\)

AC

^{2}= 6.32455532AC

^{2}= 6.32 m (to two decimal places)