Official Guide Explanation:
Problem Solving #145




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Solution and Metadata

Question: 145
Page: 81
Difficulty: 7 (Very Difficult)
Category 1: Geometry > Circles > Multiple figures
Category 2: Geometry > Triangles > Multiple figures

Explanation: Because the circles are identical, they each have the same radius. Since each side of the triangle has endpoints at two centers of the circles, the length of each side is twice the length of the radius of each circle. And because all three sides are made up this way, we know all the sides are equal, so it's an equilateral triangle.

Thus, each angle in the triangle is 60 degrees, and each side is 2r, where r is the radius of any of the circles. To find the area of the shaded region in terms of r, start with the area of the triangle. Then subtract the three sectors of the circles that are within the triangle but are not shaded.

First, the area of the triangle. The base is 2r, but the height is not one of the sides. If you draw a line parallel to the base (in this case, the top of the three sides), the result is a 30:60:90 triangle on either side, where the base is the side corresonding to the 60 degree angle. Thus, the "short" side of the triangle is r (half of the base), meaning that, using the ratio of r:r rt[3]:2r, the middle side is r rt[3].

The area of the triangle, then, is a = (1/2)bh = (1/2)(2r)(r rt[3]) = r2 rt[3].

At this point, we could assume that the 64 rt[3] term given as part of the area of the shaded region is equivalent to r2 rt[3], meaning that r = 8, and choice (B) is correct. This would be a reasonable assumption, and we'd be right.

To complete the process, however, recognize that each of the three sectors are defined by a 60 degree angle. Since the circles are identical, the 60 degree sectors are identical. So instead of calculating the area of 3 60 degree sectors, calculate one 180 degree sector--that is, the area of half of one of the circles. The area of one of the circles is π r2, so half of the area is (( π r2)/2).

Thus, the area of the shaded region is r2 rt[3] - (( π r2)/2), which must equal 64 rt[3] - 32 π :

r2 rt[3] - (( π r2)/2) = 64 rt[3] - 32 π

Again, it's easiest to assume that each term corresponds to the term on the other side that looks like it:

r2( rt[3] - (( π )/2)) = 64( rt[3] - (( π )/2))

r2 = 64

r = 8, choice (B).

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