SAT Math Multiple Choice Practice Question 88: Answer and Explanation

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Question: 88


In this diagram, where O is the center of the circle, find the measure of angle P, in degrees.

A. 32
B. 36
C. 45
D. 58
E. 64

Correct Answer: A


A. I take the liberty of naming the three unnamed points in my diagram (and you should, too):

The one angle that you know, 64°, is a central angle. Arc ST must also be 64°. Because arc RST is a semicircle, arc RS must measure 180° - 64° = 116°. Now you can look at some angles. PRT is the meeting of a tangent line (a line that touches the circle at exactly one point) and a diameter, so it must be 90°. (This fact is worth memorizing.) T is inscribed (having both its endpoints and its center on the circumference of the circle), so it must be half of its arc, 116°, and 1/2 × 116° = 58°. Finally, because triangle PRT must contain 180°, that leaves 180° - (90° + 58°) = 180° - 148° = 32° for P. (For more on tangents and inscribed angles, turn to Chapter 16.)

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