如图所示，一个底边为$16$，高为$15$的等腰三角形内有一个半圆与两条腰相切，并且半圆的直径与三角形的底边重合，问这个半圆的半径是多少？

A. $4\sqrt{3}$

B. $\frac{120}{17}$

C. 10

D. $\frac{17\sqrt{2}}{2}$

E. $\frac{17\sqrt{3}}{2}$

1. 2016 AMC 8 官方解析

翻译：一个半圆内切与一个等腰三角形，三角形的底是16，高是15，半圆的直径和等腰三角形的底重合，那么这个半圆的半径是（ ）。

等面积法$AC = \sqrt{8^2 + 15^2} = 17$

$S_{\Delta ABC} = S_{\Delta ABD} + S_{\Delta ACD}$

$16 \times 15 \div 2 = 17 \times r \div 2 \times 2$

$17r = 8 \times 15$

$r = \frac{120}{17}$.

First, we draw a line perpendicular to the base of the triangle and cut the triangle in half. The base of the resulting right triangle would be 8, and the height would be 15. Using the Pythagorean theorem, we can find the length of the hypotenuse, which would be 17. Using the two legs of the right triangle, we can find the area of the right triangle, 60. $\frac{60}{17}$ times 2 results in the radius, which is the height of the right triangle when using the hypotenuse as the base. The answer is B $\frac{120}{17}$.