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2007 AMC 10B Problem 13

Below is the professionally curated solution for Problem 13 of the 2007 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 10B solutions, or check the answer key.

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Concepts:sectorarea decompositionsymmetry

Difficulty rating: 1580

13.

Two circles of radius 22 are centered at (2,0)(2,0) and at (0,2).(0,2). What is the area of the intersection of the interiors of the two circles?

π−2\pi-2

π2\dfrac{\pi}{2}

π33\dfrac{\pi\sqrt3}{3}

2(π−2)2(\pi-2)

π\pi

Solution:

The two circles intersect at (0,0)(0,0) and (2,2).(2,2).

By symmetry, half the intersection is formed by removing an isosceles right triangle of leg length 22 from a quarter of one circle. The quarter-circle has area 14π(2)2=π\dfrac14\pi(2)^2=\pi and the triangle has area 12(2)2=2.\dfrac12(2)^2=2.

Therefore the whole region has area 2(π−2).2(\pi-2).

Thus, the correct answer is D.

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