2006 AMC 10B Problems/Problem 8

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Problem

A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle?

$\mathrm{(A) \ } 20\pi\qquad \mathrm{(B) \ } 25\pi\qquad \mathrm{(C) \ } 30\pi\qquad \mathrm{(D) \ } 40\pi\qquad \mathrm{(E) \...$

Solution

Since the area of the square is $40$ , the length of the side is $\sqrt{40}=2\sqrt{10}$ . The distance between the center of the semicircle and one of the bottom vertecies of the square is half the length of the side, which is $\sqrt{10}$ .

Using the Pythagorean Theorem to find the square of radius:

$(2\sqrt{10})^2 + (\sqrt{10})^2 = r^2$

$50=r^2$

So, the area of the semicircle is $\frac{1}{2}\cdot \pi \cdot 50 = 25\pi \Rightarrow B$

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2006 AMC 10B Problems/Problem 8

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