Problem:
Angle of is a right angle. The sides of are the diameters of semicircles as shown. The area of the semicircle on equals , and the arc of the semicircle on has length . What is the radius of the semicircle on
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Answer Choices:
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Solution:
The circle with diameter has twice the area of the corresponding semicircle; thus the area of the circle is and its radius is . Consequently . The circle with diameter has circumference , so . is the hypotenuse of the right triangle. By the Pythagorean Theorem, . Therefore , and the radius is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions