Bunuel wrote:

In the figure above, the radius of the circle with center S is twice the radius of the circle with center O, and the measure of angle RST is twice that of angle POQ. If the area of the shaded region of circle O is 3, what is the area of the shaded region of circle S?

(A) 24

(B) 12

(C) 6

(D) 3

(E) 3/2

Attachment:

2017-11-21_1004_001.png

We can let the degree measure of angle POQ = x and the radius of circle O = r. Thus the area of the shaded region of circle O = (x/360)πr^2. We are also given that this area is 3. Thus we have:

(x/360)πr^2 = 3

Furthermore, the degree measure of angle RST = 2x and the radius of circle S = 2r. Thus the area of the shaded region of circle S = (2x/360)π(2r)^2 = 2(x/360)π(4r^2) = 8(x/360)πr^2 = 8(3) = 24.

Alternate Solution:

Since the radius of the circle with center S is twice the radius of the circle with center O, the area of the circle with center S is four times the area of the circle with center O. Thus, had the angle RST been equal to the angle POQ, the area of the shaded region RST would have been 4 times the area of the shaded region POQ. However, since the angle RST is twice the angle POQ, the area RST will be twice as large, which makes it 8 times the area POQ. Since the area POQ is 3, area RST is 8 x 3 = 24.

Answer: A

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