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06 Jul 2020, 03:51
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D
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60% (02:00) correct
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Points A, B, C, and D are on the circumference of the circle centered at P, and segments BD and AC both pass through P. What is the measure of ∠BPC?
(1) \(∠APB = \frac{1}{2}∠DCP\)
(2) ∠PDC = 72°
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Pure algebraic questions like this one are no longer part of the GMAT Data Sufficiency syllabus.
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06 Jul 2020, 04:28
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Both triangles are isosceles, because in each, two sides are radii. If the two equal angles in PDC (the angles at C and D) are both y degrees, the angle at P in that triangle is 180 - 2y degrees.
In Statement 1, because we have an X-pattern at the center of the circle, angle APB and angle CPD are the same. So using the letters as defined above, Statement 1 tells us 180 - 2y = y/2, and solving we find y = 72. The angle at P is half of that from Statement 1, so is 36 degrees, and the angle we want, BPC, completes a straight line with this 36 degree angle, so is 144 degrees, and Statement 1 is sufficient.
Statement 2 tells us the angle at D is 72, so the angle at C is 72, the angle at P is 36, and again the angle we want is 144 degrees.
So the answer is D.
In Statement 1, because we have an X-pattern at the center of the circle, angle APB and angle CPD are the same. So using the letters as defined above, Statement 1 tells us 180 - 2y = y/2, and solving we find y = 72. The angle at P is half of that from Statement 1, so is 36 degrees, and the angle we want, BPC, completes a straight line with this 36 degree angle, so is 144 degrees, and Statement 1 is sufficient.
Statement 2 tells us the angle at D is 72, so the angle at C is 72, the angle at P is 36, and again the angle we want is 144 degrees.
So the answer is D.
06 Jul 2020, 04:45
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BPC = 180 - APB
= 180 - CPD
= DCP + CDP
= 2*DCP = 2*CDP
(1) APB = 1/2 * DCP
DCP = 2x = CDP
=> APB = CPD = x
=> BPC = 2*DCP = 2*2x = 4x = 4*(180/5) = 144
(2) PDC = 72 => BPC = 2*PDC = 144
Ans: D
= 180 - CPD
= DCP + CDP
= 2*DCP = 2*CDP
(1) APB = 1/2 * DCP
DCP = 2x = CDP
=> APB = CPD = x
=> BPC = 2*DCP = 2*2x = 4x = 4*(180/5) = 144
(2) PDC = 72 => BPC = 2*PDC = 144
Ans: D
