Official Guide Explanation:
Problem Solving #153




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Solution and Metadata

Question: 153
Page: 82
Difficulty: 6 (Moderately Difficult)
Category 1: Geometry > Circles > Sector/Arc
Category 2: Geometry > Triangles > Multiple figures

Explanation: Given the radius of a circle and the length of an arc, we can determine the circumference of the circle and, with that, figure out how much of the circle the arc occupies.

Since the radius is 4, the circumference is 2 π r = 8 π . The arc is ((4 π )/3), so to find the fraction of the circle that the arc constitutes:

x(8 π ) = ((4 π )/3)

x = (1/(8 π ))(((4 π )/3)) = (1/6)

One - sixth of a circle is (1/6)(360) = 60 degrees. Thus, we know that triangle RTU is an equilateral triangle. Both legs are equal (they are both equal to the radius), meaning the other two angles are equal, so all three angles must be 60.

Thus, all three sides are equal. We already know that the radius is 4, so RU must be 4 as well. Choice (D) is correct.

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