# Solution to the [[Four Quarter Circles III]] Puzzle +-- {.image} [[FourQuarterCirclesIII.jpg:pic]] > Four quarter circles. What’s the area of the orange one? =-- ## Solution by [[Congruent Triangles]] +-- {.image} [[FourQuarterCirclesIIILabelled.png:pic]] =-- 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 scale factor|area scales]] by $2$. Hence the area of the yellow quarter-circle is $2 \times 12 = 24$.