In the right triangle $ABC, AC = 12, BC = 5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?

A. $\frac{7}{6}$

B. $\frac{13}{5}$

C. $\frac{59}{18}$

D. $\frac{10}{3}$

E. $\frac{60}{13}$

1. 2017 AMC 8 官方解析

翻译：如图，直角三角形 $ABC$中，$AC = 12$，$BC = 5$，$\angle C = 90^\circ$，一个半圆形内接于直角三角形中，那么这个半圆形的半径是（ ）？

如图所示，连接$OD$、$OB$，等面积法：根据勾股定理$AB = \sqrt{5^2 + 12^2} = 13$，

$S_{\Delta ABC} = S_{\Delta AOB} + S_{\Delta BOC}$

$5 \times 12 \div 2 = 13 imes r \div 2 + r \times 5 \div 2$

$60 = 13r + 5r$

$r = \frac{10}{3}$．