# Strike a chord

Can you work out the radius of a circle from some information about a chord?

Image

The shortest distance from the chord to the centre of the circle is 3 cm.

Find the radius of the circle.

*This problem is taken from the World Mathematics Championships*

This diagram shows the information in the question.

The angle between the 3 cm line segment and the chord must be a right angle, because the 3 cm line segment joins the centre to the closest point on the chord. This means that drawing on a radius can create a right-angled triangle, as shown below:

So, by Pythagoras' Theorem, $$\begin{align}3^2+4^2&=r^2 \\

\Rightarrow 9+16&=r^2\\

\Rightarrow 25&=r^2\\

\Rightarrow 5&=r\end{align}$$

Image

The angle between the 3 cm line segment and the chord must be a right angle, because the 3 cm line segment joins the centre to the closest point on the chord. This means that drawing on a radius can create a right-angled triangle, as shown below:

Image

So, by Pythagoras' Theorem, $$\begin{align}3^2+4^2&=r^2 \\

\Rightarrow 9+16&=r^2\\

\Rightarrow 25&=r^2\\

\Rightarrow 5&=r\end{align}$$