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Geometry and Trigonometry / Circles Difficulty: Hard

The figure presents a circle with center O. Point A, lies on the upper left part of the circle. Point B lies on the upper right part of the circle. Point C is indicated at the bottom of the circle, directly below the center O. Line segment O A, line segment O B, and horizontal line segment A, B are drawn forming triangle O A, B. Vertical line segment O C is drawn.

Point O is the center of the circle above, and the measure of $angle O A, B$ is $30 degrees$ . If the length of $line segment O C$ is 18, what is the length of arc $A, B$ ?

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Explanation

Choice B is correct. Because segments OA and OB are radii of the circle centered at point O, these segments have equal lengths. Therefore, triangle AOB is an isosceles triangle, where angles OAB and OBA are congruent base angles of the triangle. It’s given that angle OAB measures $30 degrees$ . Therefore, angle OBA also measures $30 degrees$ . Let $x degrees$ represent the measure of angle AOB. Since the sum of the measures of the three angles of any triangle is $180 degrees$ , it follows that $30 degrees plus 30 degrees, plus x degrees, equals 180 degrees$ , or $60 degrees plus x degrees, equals 180 degrees$ . Subtracting $60 degrees$ from both sides of this equation yields $x degrees equals 120 degrees$ , or $the fraction 2 pi over 3$ radians. Therefore, the measure of angle AOB, and thus the measure of arc $A, B$ , is $the fraction 2 pi over 3$ radians. Since $the line segment O C$ is a radius of the given circle and its length is 18, the length of the radius of the circle is 18. Therefore, the length of arc $A, B$ can be calculated as $the fraction 2 pi over 3, end fraction, times 18$ , or $12 pi$ .

Choices A, C, and D are incorrect and may result from conceptual or computational errors.