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Mathematics > Standard Multiple Choice

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The circle shown above has center O and a radius of length 5. If the area of the shaded region is 20 times pi, what is the value of x?

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In order to find the value of x, you should first determine the measure of the angle that is located at point O in the right triangle. To determine this angle, you must calculate what fraction of the circle’s area is unshaded. The radius r of the circle is 5 and its area is pi times r^2, or 25 times pi. The area of the shaded region is 20 times pi, so the area of the unshaded region must be 5 times pi. Therefore, the fraction of the circle’s area that is unshaded is (5 times pi) over (25 times pi), or 1 over 5. A circle contains a total of  three hundred and sixty degrees of arc, which means that 1 over 5 of  360 degrees, or 72 degrees, is the measure of the angle at point O in the unshaded region. Since you now know that two of the three angles in the triangle measure 72 degrees and 90 degrees and that the sum of the measures of the three angles is always 180 degrees, the third angle must measure 18 degrees. Therefore, x = 18.

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