Official Guide Explanation:
Problem Solving #162




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Solution and Metadata

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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