Overlapping semicircles
Two semicircles overlap, can you find this length?
Age
14 to 16
Challenge level
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.