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05 Feb 2019, 03:06
Kudos
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
73% (03:11) correct
27%
(02:49)
wrong
based on 30
sessions
History
Date
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Not Attempted Yet
In the circle with center O above, PS = 8. If x = 75, then what is the perimeter of the shaded region?
A. \(6-2\sqrt{3}\)
B. \(\frac{2\pi}{3}\)
C. \(\frac{2\pi}{3}+8\)
D. \(\frac{2\pi}{3}-2\sqrt{3}+6\)
E. \(\frac{2\pi}{3}+2\sqrt{3}+6\)
Attachment:
2019-02-05_1402.png [ 32.6 KiB | Viewed 1364 times ]
Bunuel
IMO D
x=75, therefore \(\angle QOT=180-75-75=30\)
Triangle OQT is a 30-60-90 triangle, and with radius = OQ = 4 we have, QT=2 and OT = \(2\sqrt3\)
Thus, TR = OR-OT= \(4-2\sqrt3\)
Also arc QR \(= \frac{30}{360} * 2\pi*4 = \frac{2\pi}{3}\)
And perimeter of shaded region = QT+TR+arc QR = \(2+4-2\sqrt3+\frac{2\pi}{3}\)
\(=\frac{2\pi}{3}-2\sqrt3+6\)
---------------------------------------------------------
Kudos if helpful!
Attachments
2019-02-05_1402.png [ 33.97 KiB | Viewed 1164 times ]
05 Feb 2019, 04:02
Kudos
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Bunuel
PS = 8 so radius is 4
Since x = 75, x + angle QOR + x = 180
angle QOR = 30 degrees
Note that we get a 30-60-90 triangle (non shaded sector) in which OQ = 4 (radius)
The ratio of sides is \(1:\sqrt{3}:2\) in which 2 corresponds to a value of 4 (hypotenuse)
So the side opposite 30 degrees angle is 2 and the side opposite 60 degrees angle is \(2*\sqrt{3}\).
Length of QR will be (30/360) * Circumference of circle \(= (1/12) * 2*\pi*4 = \frac{2}{3} *\pi\)
Perimeter of shaded region \(= \frac{2}{3}*\pi + 2 + (4 - 2*\sqrt{3}) = \frac{2}{3}*\pi + 6 - 2*\sqrt{3}\)
Answer (D)
