Circular Routes

A relatively simple problem with a nice solution.

The upper route is a semicircle with diameter 20 therefore it has length \(10\pi\).

The lower route is made of semicircles with lengths \(2.5\pi\), \(5\pi\) and \(2.5\pi\). which sum to \(10\pi\).

The routes therefore have the same length.

If the diameters of the semicircles are \(x_i\) then the arc lengths are \(\frac{1}{2}\pi x_i\). The route has length \(\Sigma \frac{1}{2} \pi x_i\) and using the fact that \(\Sigma x_i=AB\), the lower route has length \(\frac{1}{2} \pi AB\) which is the length of the upper route.