Question

半径为 $5$ 个单位的三段圆弧围成了图示区域。弧 $AB$ 和 $AD$ 是四分之一圆，弧 $BCD$ 是半圆。这个区域的面积是多少平方单位？

Three circular arcs of radius $5$ units bound the region shown. Arcs $AB$ and $AD$ are quarter-circles, and arc $BCD$ is a semicircle. What is the area, in square units, of the region?

$25$

$10+5\pi$

$50$

$50+5\pi$

$25\pi$

答案：C

知识点：面积分割圆面积

难度评级：1460

解答：

如下图，作一个覆盖图形下半部分的矩形。

于是 $[ABCD] = [ABD] + [BCD].$

还知道 $[ABD] = [BDEF] - [ABE]$ $- [ADF].$

$ABE$ 和 $ADF$ 都是四分之一圆，它们合起来形成一个半圆，面积与 $BCD$ 相同。

因此 $[ABD] = [BDEF] - [BCD]$ 并且 $[ABCD] = [BCD] + [BDEF]$ $- [BCD]$ $= [BDEF] = 10 \cdot 5 = 50.$

所以正确答案是 C。

Create a rectangle that covers the bottom half of the figure as shown below.

Then, we get that $[ABCD] = [ABD] + [BCD].$

We also know that $[ABD] = [BDEF] - [ABE]$ $- [ADF].$

$ABE$ and $ADF$ are both quarter-circles that form a semicircle with the same area as $BCD.$

This means that $[ABD] = [BDEF] - [BCD]$ and $[ABCD] = [BCD] + [BDEF]$ $- [BCD]$ $= [BDEF] = 10 \cdot 5 = 50.$

Thus, C is the correct answer.

2000 AMC 8 第 19 题

答案：C

难度评级：1460

解答：

其他年份的第 19 题