In the figure, the three circles are concentric, and the area of the smallest circle coincides with the area of the largest ring, highlighted in gray. The radius of the smallest circle is $5 \mathrm{~cm}$ and of the largest $13 \mathrm{~cm}$. What is the radius (in cm) of the intermediate circle?
(a) 12
(c) $10 \sqrt{65}$
(e) $12 \sqrt{2}$
(b) 11
(d) $5 \sqrt{3}$

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-013.jpg?height=402&width=403&top_left_y=250&top_left_x=1472)
The correct option is (a).

The area of the largest circle is $13^{2} \pi=169 \pi$ and that of the smallest is $5^{2} \pi=25 \pi$, which is also the area of the largest ring. Let $r$ be the radius of the intermediate circle. Then, the area of the largest ring is $169 \pi-\pi r^{2}$. Therefore, $169 \pi-\pi r^{2}=25 \pi$, that is, $\pi r^{2}=169 \pi-25 \pi=144 \pi$, from which $r^{2}=144$ and $r=12 \text{ cm}$.