Official Guide Explanation:
Data Sufficiency #160

Background

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Solution and Metadata

Question: 160
Page: 287
Difficulty: 6 (Moderately Difficult)
Category 1: Geometry > Circles > Multiple figures
Category 2: Geometry > Circles > Sector/Arc
Category 3: Geometry > Triangles > Multiple figures

Explanation: To find the circumference of the circle, you need the radius, or something that can give you the radius. Statement (1) is sufficient: the triangle has angles in a ratio of 45:45:90, which means that the sides have a ratio of x:x:x rt[2]. So, if the perimeter of the triangle is 20 + 10 rt[2], x + x + x rt[2] = 20 + 10 rt[2]. You can solve for x, which is also a radius of the circle, so you can solve for the cirumference of the circle.

Statement (2) is also sufficient: arc XYZ is (1/4) the circumference of the entire circle, since it is defined by an angle of 90, which is one - quarter of 360, the angle measure of the entire circle. If 5 π is a quarter of the circumference, the circumference must be 4(5 π ) = 20 π . Choice (D) is correct.

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