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Mathematics>Standard Multiple Choice

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

In the figure above, a shaded circle with diameter line C D is tangent to a large semicircle with diameter line A B at points C and D. Point C is the center of the semicircle, and line A B is perpendicular to line C D. If the area of the large semicircle is 24, what is the area of the shaded circle?

(A) 8
(B) 10
(C) 12
(D) 14
(E) It cannot be determined from the information given.

答案和解析

答案：C

解析：

Segment line C D is a radius of the large semicircle, and it is also a diameter of the shaded circle, so the radius of the large semicircle is equal to twice the radius of the shaded circle. Let r be the radius of the shaded circle. Then 2*r is the radius of the large semicircle, and so the area of the semicircle is (pi*(2*r)^2)/2 = (4*pi*r^2)/2 = 2*pi*r^2. This area is given as 24, and so the area of the shaded circle is (pi*r^2) = (2*pi*r^2)/2 = 24/2 = 12.