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Geometry and Trigonometry / Circles Difficulty: Medium

Points $M$ , $N$ , and $P$ lie on the circle shown. On this circle, minor arc $M N$ has a length of $39$ centimeters and major arc $M P N$ has a length of $195$ centimeters. What is the circumference, in centimeters, of the circle shown?

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Explanation

Choice D is correct. Since the endpoints of minor arc $M N$ and major arc $M P N$ are the same, and the arcs together form a full circle, the sum of the lengths of these two arcs is equal to the circumference of the circle. It's given that the length of minor arc $M N$ is $39$ centimeters and the length of major arc $M P N$ is $195$ centimeters. Therefore, the circumference of the circle, in centimeters, is $39 plus 195$ , or $234$ .

Choice A is incorrect. This is the length, in centimeters, of minor arc $M N$ , not the circumference, in centimeters, of the circle.

Choice B is incorrect. This is the difference of the lengths of major arc $M P N$ and minor arc $M N$ , in centimeters.

Choice C is incorrect. This is the length, in centimeters, of major arc $M P N$ , not the circumference, in centimeters, of the circle.