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22 Jul 2014, 07:34
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A
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Difficulty:
25%
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Question Stats:
78% (02:17) correct
22%
(02:25)
wrong
based on 751
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If P, Q and R are the centers of circles P, Q, and R and the points P, Q, R and T all lie on the same line, what portion of circle P is shaded?
A. 3/16
B. 1/5
C. 6/25
D. 1/4
E. 3/8
Hi, I want to know why we don't get same answer using PQ = 2?
OE
Let us say that line segment RT has a length of 1. RT is the radius of circle R, so circle R has a radius of 1.
Line segment QT is the diameter of circle R, so it has a length of 2 (twice the radius of circle R). Segment QT also happens to be the radius of circle Q, which therefore has a radius of 2.
Line segment PT, being the diameter of circle Q, has a length of 4. Segment PT also happens to be the radius of circle P, which therefore has a radius of 4.
The question is asking us what fraction of circle P is shaded. The answer will be
(shaded area) ÷ (area of circle P)
The area of circle P is π(4)2, which equals 16π. The shaded area is just the area of circle Q (i.e. π(2)^2, which equals 4π) minus the area of circle R (i.e. π(1)^2, which equals π). Therefore, the answer to our question is
(4π - π) / 16π
Line segment QT is the diameter of circle R, so it has a length of 2 (twice the radius of circle R). Segment QT also happens to be the radius of circle Q, which therefore has a radius of 2.
Line segment PT, being the diameter of circle Q, has a length of 4. Segment PT also happens to be the radius of circle P, which therefore has a radius of 4.
The question is asking us what fraction of circle P is shaded. The answer will be
(shaded area) ÷ (area of circle P)
The area of circle P is π(4)2, which equals 16π. The shaded area is just the area of circle Q (i.e. π(2)^2, which equals 4π) minus the area of circle R (i.e. π(1)^2, which equals π). Therefore, the answer to our question is
(4π - π) / 16π
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22 Jul 2014, 07:51
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goodyear2013
The radius of P is twice the radius of Q, which is twice the radius of R.
Say the radius of R is 1, so the radius of Q is 2 and the radius of P is 4.
The area of the shaded region = area of Q - area of R = \(4\pi-\pi=3\pi\).
The area of P = \(16\pi\).
The ratio = 3/16.
Answer: A.
As for your question PR is not the radius of any circle, so assuming a value for it is not a good idea. PR = radius of Q + radius of R = 2x + x = 2 --> radius of R = x = 2/3 --> the radius of Q is 4/3 --> the radius of P is 8/3.
The area of the shaded region = area of Q - area of R = \(\frac{16}{9}\pi-\frac{4}{9}\pi=\frac{12}{9}\pi\).
The area of P = \(\frac{64}{9}\pi\).
The ratio = 12/64 = 3/16. Thew same answer.
General Discussion
22 Jul 2014, 08:09
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goodyear2013
PQ is the radius of the middle circle, Q. In my solution above I assumed exactly that PQ = QT = 2.
