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Subject Test Math 1

Question 31 of 32

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SAT Subject Test

In the figure, rectangle A B C D is inscribed in a circle; that is, the vertices A, B, C, and D lie on the circle. The rectangle A B C D partitions the circle into 5 regions, the rectangular region and four other non-overlapping regions outside the rectangle. The four non overlapping regions outside the rectangle and inside the circle are shaded.

Rectangle A B C D is inscribed in the circle shown above. If the length of side stack A B with bar on top is 5 and the length of side stack B C with bar on top is 12, what is the area of the shaded region?

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40.8

Correct Answer: 
No

53.1

Correct Answer: 
No

72.7

Correct Answer: 
Yes

78.5

Correct Answer: 
No

81.7

Correct Answer: 
No

The correct answer is C.

The area of the shaded region can be found by subtracting the area of rectangle A B C D from the area of the circle. To determine the area of the circle, first find the radius r, and then compute the area pi r to the power of 2 end exponent. Since rectangle A B C D is inscribed in the circle, angle A B C is an inscribed right angle, and thus stack A C with bar on top is a diameter of the circle. Applying the Pythagorean theorem to right triangle A B C,one finds the length of side stack A C with bar on top is square root of 5 to the power of 2 end exponent plus 12 to the power of 2 end exponent end root equals square root of 169 end root equals 13 Thus the radius of the circle is fraction numerator 13 over denominator 2 end fraction, and the area of the circle is pi left parenthesis fraction numerator 13 over denominator 2 end fraction right parenthesis to the power of 2 end exponent equals fraction numerator 169 over denominator 4 end fraction pi. The area of rectangle A B C D is 5 cross times 12 equals 60, and therefore, the area of the shaded region is fraction numerator 169 over denominator 4 end fraction pi minus 60 almost equal to 72.7 invisible times.

Question Difficulty: 
hard