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2006 AMC 10B Problem 8

Problem 8 of 25EasierGeometry

A square of area 4040 is inscribed in a semicircle as shown. What is the area of the semicircle?

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Solution

Let the square have side s,s, so s2=40.s^2=40. Its base lies centered on the diameter, and a top corner at (s2,s)\left(\tfrac{s}{2},s\right) lies on the circle. Then r2=(s2)2+s2=404+40=50.r^2=\left(\tfrac{s}{2}\right)^2+s^2=\tfrac{40}{4}+40=50. The semicircle area is 12πr2=12π(50)=25π.\tfrac12\pi r^2=\tfrac12\pi(50)=25\pi. Thus, the correct answer is B.

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Concepts: circle area · Pythagorean Theorem · square (geometry)

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.