Notes
four quarter circles iii solution

Solution to the Four Quarter Circles III Puzzle

Solution by Congruent Triangles

Consider the diagram labelled as above. Triangle $O D C$ is an isosceles right-angled triangle as it comes from the quarter circle, therefore angle $D \hat{O} C$ is $45^\circ$. Angle $C \hat{O} B$ is then $90^\circ - 45^\circ = 45^\circ$, so then triangles $C O A$ and $C O B$ are such that they have the same angle at $O$ with adjacent sides of equal length. Therefore they are congruent.

This means that line segments $C B$ and $A C$ have the same length. As triangle $B Q C$ is also an isosceles right-angled triangle, $B C$ has length $\sqrt{2}$ times that of $B Q$. Therefore, the yellow quarter-circle is $\sqrt{2}$ times larger than the red quarter-circle. Its area scales by $2$.

Hence the area of the yellow quarter-circle is $2 \times 12 = 24$.