15. A fragment of an ancient circular plate, now broken, has been found, in the shape shown in the figure. $C$ is the midpoint of the segment $A B$, while $D$ is the midpoint of the arc $A B$. Knowing that $A B$ measures $24 \mathrm{~cm}$ and $C D$ measures $6 \mathrm{~cm}$, by how many $\mathrm{cm}$ was the radius of the original plate?

![](https://cdn.mathpix.com/cropped/2024_04_17_59c98a7c6326a6aa0f66g-2.jpg?height=205&width=217&top_left_y=681&top_left_x=1182)
(A) 16
(B) 12
(C) 18
(D) 20

(E) 15

![](https://cdn.mathpix.com/cropped/2024_04_17_59c98a7c6326a6aa0f66g-2.jpg?height=217&width=339&top_left_y=1372&top_left_x=1061)
Problem 15. The answer is (E).

Let $O$ be the center of the circular plate and $x=\overline{O C}$. The radius of the circle is thus $r=x+6$, and by the Pythagorean theorem, we have $(x+6)^{2}=r^{2}=x^{2}+(24 / 2)^{2}$, from which $x=9$ and $r=15$.

(Problem proposed by F. Getman)