AMC10 2007 B

AMC10 2007 B · Q13

AMC10 2007 B · Q13. It mainly tests Circle theorems, Area & perimeter.

Two circles of radius 2 are centered at (2, 0) and at (0, 2). What is the area of the intersection of the interiors of the two circles?

两个半径为2的圆分别以(2, 0)和(0, 2)为圆心。两个圆内部交集的面积是多少？

(A) $\pi - 2$ $\pi - 2$

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

(D) $2(\pi - 2)$ $2(\pi - 2)$

(E) $\pi$ $\pi$

Answer

Correct choice: (D)

正确答案：（D）

Solution

The two circles intersect at (0, 0) and (2, 2), as shown. Half of the region described is formed by removing an isosceles right triangle of leg length 2 from a quarter of one of the circles. Because the quarter-circle has area (1/4)$\pi$(2)$^2$ = $\pi$ and the triangle has area (1/2)(2)$^2$ = 2, the area of the region is 2($\pi$ − 2).

两个圆相交于(0, 0)和(2, 2)，如图所示。所述区域的一半是由从一个圆的一个四分之一中减去腿长为2的等腰直角三角形形成的。因为四分之一圆的面积为$(1/4)\pi(2)^2$ = $\pi$，三角形的面积为$(1/2)(2)^2$ = 2，所以该区域的面积为$2(\pi$ − 2)。

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