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## Official Guide Explanation:

Data Sufficiency #173

**Background**

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Review (12th ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

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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:

2^{2} + a^{2} = x^{2}

2^{2} + b^{2} = y^{2}

x^{2} + y^{2} = (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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