# Overlapping Semicircles

Two semicircles overlap, can you find this length?

## Problem

Image

Two semicircles are drawn in a rectangle as shown.

What is the width of the overlap of the two semicircles?

This problem is taken from the UKMT Mathematical Challenges.

## Student Solutions

In the diagram below, only half of the rectangle is shown - so the height is 4 cm.

Since the length of the rectangle is 10 cm, the radius of the semicircle is 5 cm.

Image

The red triangle is right-angled, with hypotenuse 5 cm (since it is the radius of the circle).

Applying Pythagoras' Theorem to find $x,$ $$\begin{align}x^2+4^2&=5^2\\

\Rightarrow x^2+16&=25\\

\Rightarrow x^2&=9\\

\Rightarrow x&=3\end{align}$$

So the width of the overlap is $2\times3=6$ cm.