PSAT Math Multiple-Choice Question 87: Answer and Explanation

Question: 87

In the figure above, O is the center of the circle, the radius of the circle is x, and the length of minor arc PQ is . What is the area of sector POQ?

• A.
• B.
• C.
• D.

Correct Answer: A

Explanation:

A

The question asks for the area of a sector of the circle. There are variables in the answer choices, so plug in. Plug in x = 3. Write down the equations needed answer the question. The equation for area of a circle is A = πr², so the area of the circle is π (3) = 9π. The equation for circumference is C = 2πr, so the circumference of the circle is 2π(3) = 6π. The length of arc PQ is . The parts of a circle have a proportional relationship. In this circle, the fraction of the arc length out of the total circumference is the same as the fraction of the sector area out of the total area. Set up the proportion , then plug in the given information to get . Divide the fraction on the left side of the equation to get . Cross-multiply to get 9π = 36(sector PQR), then divide both sides by 36 to get sector PQR = . This is the target value; circle it. Now plug x = 3 into the answer choices to see which one matches the target value. Choice (A) becomes . Keep (A), but check (B), (C), and (D) just in case. Choice (B) becomes . This does not match the target, so eliminate (B). Choice (C) becomes . Eliminate (C). Choice (D) becomes . Eliminate (D). The correct answer is (A).