Notes
semi-circle in a circle solution

Solution to the Semi-Circle in a Circle Puzzle

Solution by Intersecting Chords Theorem

In the diagram above, let $r$ be the radius of the inner semi-circle. So $r$ is the length of each of $E A$, $E B$, and $E F$. The length of $D E$ is then $6 + r$ so applying the intersecting chords theorem to $A B$ and $C D$ results in:

This has solution $r = 3$ (the other possible solution is $r = -2$ but as $r$ is a length this isn’t possible).

The diameter of the outer circle is $7 + r = 10$ so its area is $25 \pi$. The area of the inner semi-circle is $\frac{9}{2} \pi$. Therefore $\frac{41}{50}$ths of the circle is shaded.