Official Guide Explanation:
Data Sufficiency #173




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Solution and Metadata

Question: 173
Page: 288
Difficulty: 7 (Very Difficult)
Category 1: Geometry > Circles > Multiple figures
Category 2: Geometry > Triangles > Pythag
Category 3: Geometry > Triangles > Multiple figures

Explanation: It's important for the purposes of this problem to know that, in a semicircle, an angle created by the intersection of the circumference of the circle and two lines that also go through the edges of the diameter is a right angle. In other words, angle PQR is a right angle. So, there are 3 right triangles in the diagram. If we call PQ x and QR y, the equations are as follows:

22 + a2 = x2

22 + b2 = y2

x2 + y2 = (a + b)2

In short: three equations and four variables. To solve, we need a fourth equation.

Statement (1) offers that: if a = 4, we can plug 4 into each of the equations, reducing them to three equations with three variables. It would take some time to solve, but it's enough to know that we can.

Statement (2) works the same way: since b = 1, we can plug 1 into each of the equations and solve. Choice (D) is correct.

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