SAT Math Multiple Choice Practice Question 55: Answer and Explanation

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Question: 55


Find the shaded area in this figure, where O is the center of the circle.

A. 676π - 240
B. 676π - 120
C. 576π - 120
D. 169π - 240
E. 169π - 120

Correct Answer: E


E. LM is a diameter, which means it divides the circle into two 180° arcs. Because N is an inscribed angle (it's on the edge of the circle), its measure is half of the lower 180° arc, or 90°. Yep, it's a right triangle! Because you know the lengths of two sides of the triangle, you can find the third. Remember that 26 is the hypotenuse, so your equation should be x2 + 242 = 262, which gives you x2 + 576 = 676, x2 = 100, and x = 10. Now for the real problem — finding the area of the shaded portion of the circle, which equals the area of the circle minus the area of the triangle. The circle's diameter is 26, so its radius is 13, and the area of a circle = πr2 = π(13)2 = 169π. The triangle's area is 1/2 × base × height, but you need to be careful. Twenty-six isn't the base because you don't have a height to go with it. Instead, you need to use the two legs of the right triangle as the base and height. (It may help to rotate the book if you have trouble visualizing this concept.) So the triangle's area is 1/2(10)(24) = 120, and the answer is 169π - 120.

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