Point O is the center of the circle above, and the measure of is . If the length of is

Question

Point O is the center of the circle above, and the measure of is . If the length of is 18, what is the length of arc ?

Accepted Answer

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&rsquo;s given that angle OAB measures . Therefore, angle OBA also measures . Let represent the measure of angle AOB . Since the sum of the measures of the three angles of any triangle is , it follows that , or . Subtracting from both sides of this equation yields , or radians. Therefore, the measure of angle AOB, and thus the measure of arc , is radians. Since 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 can be calculated as , or . Choices A, C, and D are incorrect and may result from conceptual or computational errors.

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