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## Official Guide Explanation:Problem Solving #162

Background

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Quantitative Review (2nd 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.

Question: 162
Page: 83
Difficulty: 6 (Moderately Difficult)
Category 1: Geometry > Circles > Multiple figures
Category 2: Geometry > Triangles > Multiple figures

Explanation: The equation OC = AC = AB tells us a lot. First, that ABC is an isoceles triangle. Next, that AOC is also an isoceles triangle. Not only is the marked angle equal to x, but CAO is equal to x as well.

To express some of the other things we know, call angle ABO y. Since AB = BC, angle ACB is also y. And because OA = OB (both are radiuses), angle AOB must be y as well. Thus, in the triangle AOB, we know that y + y + x = 180, or x = 180 - 2y.

Finally, let's express all three angles in triangle AOC in terms of y. We know that AOC and CAO are both x, so they are 180 - 2y. We know that ACB is y, so because that angle and ACO form a straight line, ACO is 180 - y. Those three angles must sum to 180, so we can solve for y:

180 = 2(180 - 2y) + 180 - y

180 = 360 - 4y + 180 - y

5y = 360

y = 72

Now, since x = 180 - 2y, we can solve:

x = 180 - 2(72) = 180 - 144 = 36, choice (B).

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