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Mathematics>Standard Multiple Choice

Read the following SAT test question and then click on a button to select your answer.

The circle shown above has center O and a radius of length 5. If the area of the shaded region is 2pi, what is the value of x?

(A) 18
(B) 36
(C) 45
(D) 54
(E) 72

答案和解析

答案：A

解析：

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 pir^2, or 25pi. The area of the shaded region is 20pi, so the area of the unshaded region must be 5pi. Therefore, the fraction of the circle’s area that is unshaded is (5pi) / (25pi), or 1 / 5. A circle contains a total of three hundred and sixty degrees of arc, which means that 1 / 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.