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Geometry and Trigonometry / Circles Difficulty: Hard

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 $A, B C$ ?

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Explanation

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.