2010 AMC 8 Problems/Problem 23
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Semicircles and pass through the center . What is the ratio of the combined areas of the two semicircles to the area of circle ?
Solution
According to the pythagorean theorem, The radius of the larger circle is:
Therefore the area of the larger circle is:
Using the coordinate plane given we find that the radius of the two semicircles to be 1. Therefore the area of the two semicircles is:
Finally the ratio of the combined areas of the two semicircles to the area of circle is .