AMC8 2013

AMC8 2013 · Q23

AMC8 2013 · Q23. It mainly tests Pythagorean theorem, Circle theorems.

Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$, and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$. What is the radius of the semicircle on $\overline{BC}$?

$\triangle ABC$ 的 $\angle ABC$ 是直角。$\triangle ABC$ 的边是示意中的半圆的直径。半圆 $\overline{AB}$ 的面积等于 $8\pi$，半圆 $\overline{AC}$ 的弧长为 $8.5\pi$。半圆 $\overline{BC}$ 的半径是多少？

(A) 7 7

(B) 7.5 7.5

(D) 8.5 8.5

(E) 9 9

Answer

Correct choice: (B)

正确答案：（B）

Solution

If the semicircle on $\overline{AB}$ were a full circle, the area would be $16\pi$. $\pi r^2=16 \pi \Rightarrow r^2=16 \Rightarrow r=+4$, therefore the diameter of the first circle is $8$. The arc of the largest semicircle is $8.5 \pi$, so if it were a full circle, the circumference would be $17 \pi$. So the $\text{diameter}=17$. By the Pythagorean theorem, the other side has length $15$, so the radius is $\boxed{\textbf{(B)}\ 7.5}$

如果 $\overline{AB}$ 上的半圆是全圆，面积将是 $16\pi$。 $\pi r^2=16 \pi \Rightarrow r^2=16 \Rightarrow r=+4$，因此第一个圆的直径是 8。最大半圆的弧长是 $8.5 \pi$，如果是全圆，周长将是 $17 \pi$。所以直径 = 17。根据勾股定理，另一边长为 15，因此半径是 $\boxed{\textbf{(B)}\ 7.5}$

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