2009 AMC 10A Problem 6

Problem 6 of 25EasierGeometry

A circle of radius

2

is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle’s area is shaded?

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Solution

The inscribed circle rests on the diameter and is tangent to the arc, so the semicircle has radius

4.

Its area is

\dfrac12 \pi (4)^2 = 8\pi.

The circle’s area is

\pi(2)^2 = 4\pi,

so the shaded area is

8\pi - 4\pi = 4\pi.

The shaded fraction is

\dfrac{4\pi}{8\pi} = \dfrac12.

Thus, the correct answer is A.

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