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Test
Math
Domain
Geometry and Trigonometry
Skill
Circles
Difficulty
Hard
ID: c8345903
Modded SAT Question Bank by Abdullah Mallik
The figure presents a circle with center O. There are four points on the circle. Going clockwise around the circle, the four points are A, B, C, and D.  Points A and C divide the circle into two arcs, arc A, D C and arc A B C. The length of arc A D C is less than the length of arc A, B C. Radius O A and radius O C are drawn. Angle A O C, the central angle corresponding to arc A, D C, measures x degrees

The circle above has center O, the length of arc A, D C is 5 pi, and x equals 100. What is the length of arc A, B C ?

  1. 9 pi

  2. 13 pi

  3. 18 pi

  4. 13 halves pi


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Correct Answer: B
Rationale

Choice B is correct. The ratio of the lengths of two arcs of a circle is equal to the ratio of the measures of the central angles that subtend the arcs. It’s given that arc A D C is subtended by a central angle with measure 100°. Since the sum of the measures of the angles about a point is 360°, it follows that arc A B C is subtended by a central angle with measure 360 degrees minus 100 degrees, equals 260 degrees. If s is the length of arc A B C, then s must satisfy the ratio the fraction s over 5 pi, end fraction equals, the fraction 260 over 100. Reducing the fraction 260 over 100 to its simplest form gives the fraction 13 over 5. Therefore, the fraction s over 5 pi, end fraction, equals, the fraction 13 over 5. Multiplying both sides of the fraction s over 5 pi, end fraction, equals, the fraction 13 over 5 by 5 pi yields s equals 13 pi.

Choice A is incorrect. This is the length of an arc consisting of exactly half of the circle, but arc A B C is greater than half of the circle. Choice C is incorrect. This is the total circumference of the circle. Choice D is incorrect. This is half the length of arc A B C, not its full length.

Question Difficulty: Hard
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