In the diagram, $Z$ lies on $X Y$ and the three circles have diameters $X Z, Z Y$ and $X Y$. If $X Z=12$ and $Z Y=8$, then the ratio of the area of the shaded region to the area of the unshaded region is
(A) $12: 25$
(B) $12: 13$
(C) $1: 1$
(D) $1: 2$
(E) $2: 3$

![](https://cdn.mathpix.com/cropped/2024_04_20_ac36362783317e0251fdg-093.jpg?height=383&width=447&top_left_y=1928&top_left_x=1294)
The circle with diameter $X Z=12$ has radius $\frac{1}{2}(12)=6$ so has area $\pi\left(6^{2}\right)=36 \pi$.

The circle with diameter $Z Y=8$ has radius $\frac{1}{2}(8)=4$ so has area $\pi\left(4^{2}\right)=16 \pi$.

Thus, the total unshaded area is $36 \pi+16 \pi=52 \pi$.

Since $X Z Y$ is a straight line, then $X Y=X Z+Z Y=12+8=20$.

The circle with diameter $X Y=20$ has radius $\frac{1}{2}(20)=10$, so has area $\pi\left(10^{2}\right)=100 \pi$.

The shaded area equals the area of the circle with diameter $X Y$ minus the unshaded area, or $100 \pi-52 \pi=48 \pi$.

Therefore, the ratio of the area of the shaded region to the area of the unshaded region is $48 \pi: 52 \pi$ or $48: 52$ or $12: 13$.

ANswer: (B)