AMC10 2003 B

AMC10 2003 B · Q19

AMC10 2003 B · Q19. It mainly tests Circle theorems, Area & perimeter.

Three semicircles of radius 1 are constructed on diameter AB of a semicircle of radius 2. The centers of the small semicircles divide AB into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? [Diagram shows large semicircle radius 2 on diameter AB, with three small semicircles of radius 1 inside it along AB]

在大半径为2的半圆的直径AB上，构造了三个半径为1的半圆。小半圆的圆心将AB分成四个等长的线段，如图所示。阴影区域是大半圆内但在小半圆外的面积是多少？[图示：在直径AB上的大半圆半径2，沿AB在其内部有三个半径1的小半圆]

(A) π − √3 π − √3

(B) π − √2 π − √2

(D) π + √3 / 2 π + √3 / 2

(E) 7/6 π − √3 / 2 7/6 π − √3 / 2

Answer

Correct choice: (E)

正确答案：（E）

Solution

(E) The area of the larger semicircle is $\frac{1}{2}\pi(2)^2=2\pi.$ The region deleted from the larger semicircle consists of five congruent sectors and two equilateral triangles. The area of each of the sectors is $\frac{1}{6}\pi(1)^2=\frac{\pi}{6}$ and the area of each triangle is $\frac{1}{2}\cdot 1\cdot\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4},$ so the area of the shaded region is $2\pi-5\cdot\frac{\pi}{6}-2\cdot\frac{\sqrt{3}}{4}=\frac{7}{6}\pi-\frac{\sqrt{3}}{2}.$

（E）较大半圆的面积为 $\frac{1}{2}\pi(2)^2=2\pi.$ 从较大半圆中删去的区域由五个全等扇形和两个等边三角形组成。每个扇形的面积为 $\frac{1}{6}\pi(1)^2=\frac{\pi}{6}$ 每个三角形的面积为 $\frac{1}{2}\cdot 1\cdot\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4},$ 因此阴影部分的面积为 $2\pi-5\cdot\frac{\pi}{6}-2\cdot\frac{\sqrt{3}}{4}=\frac{7}{6}\pi-\frac{\sqrt{3}}{2}.$

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