AMC8 2013

AMC8 2013 · Q20

AMC8 2013 · Q20. It mainly tests Pythagorean theorem.

A $1 \times 2$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?

一个 $1 \times 2$ 的矩形内接于一个半圆中，长边在直径上。这个半圆的面积是多少？

(A) \frac{\pi}{2} \frac{\pi}{2}

(B) \frac{2\pi}{3} \frac{2\pi}{3}

(D) \frac{4\pi}{3} \frac{4\pi}{3}

(E) \frac{5\pi}{3} \frac{5\pi}{3}

Answer

Correct choice: (C)

正确答案：（C）

Solution

A semicircle has symmetry, so the center is exactly at the midpoint of the 2 side on the rectangle, making the radius, by the Pythagorean Theorem, $\sqrt{1^2+1^2}=\sqrt{2}$. The area is $\frac{2\pi}{2}=\boxed{\textbf{(C)}\ \pi}$.

半圆具有对称性，所以圆心正好在矩形 2 边的中点处，根据勾股定理，半径为 $\sqrt{1^2+1^2}=\sqrt{2}$。面积为 $\frac{2\pi}{2}=\boxed{\textbf{(C)}\ \pi}$。

Topics

GE.005 Pythagorean theorem