AMC 8 2010 Problem 23

AMC8 2010 年第 23 题

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Semicircles POQ and ROS pass through the center of circle O . What is the ratio of the combined areas of the two semicircles to the area of circle O ?

半圆 POQ 和 ROS 通过圆的圆心。则这两个半圆的面积总和与圆的面积的比值是多少？

A. $\displaystyle \frac{\sqrt 2}{4}$
B. $\displaystyle \frac{1}{2}$
C. $\displaystyle \frac{2}{\pi}$
D. $\displaystyle \frac{2}{3}$
E. $\displaystyle \frac{\sqrt 2}{2}$

正确答案 · Correct Answer

解析 · Solution

Solution 1

By the Pythagorean Theorem, the radius of the larger circle turns out to be

\sqrt{1^2 + 1^2} = \sqrt{2}

. Therefore, the area of the larger circle is

(\sqrt{2})^2\pi = 2\pi

. Using the coordinate plane given, we find that the radius of each of the two semicircles to be 1. So, the area of the two semicircles is

1^2\pi=\pi

. Finally, the ratio of the combined areas of the two semicircles to the area of circle

O

\displaystyle \frac{1}{2}

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