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O is the center of the circle. What is <PRQ?

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O is the center of the circle. What is <PRQ?  [#permalink]

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New post Updated on: 03 Sep 2012, 01:43
1
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00:00
A
B
C
D
E

Difficulty:

  15% (low)

Question Stats:

83% (01:18) correct 17% (01:48) wrong based on 64 sessions

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O is the center of the circle. What is <PRQ?

A. 10
B. 20
C. 30
D. 60
E. 75

Originally posted by harikris on 03 Sep 2012, 00:15.
Last edited by Bunuel on 03 Sep 2012, 01:43, edited 2 times in total.
Renamed the topic and edited the question.
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Re: Doubt  [#permalink]

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New post 03 Sep 2012, 00:33
PO is the radius of the circle.
QO is the radius of the circle.
We already know that triangle POQ is isosceles.
Angle PQO is 60 degrees. Because the triangle is isosceles, angle OPQ is also 60 degrees.
Angle PQR is 90 degrees (triangle inscribed in a semicircle).

180 - PQR - RPQ = QRP
180 - 90 - 60 = QPR = 30

Answer C.
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Re: O is the center of the circle. What is <PRQ?  [#permalink]

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New post 03 Sep 2012, 01:59
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O is the center of the circle. What is <PRQ?

A. 10
B. 20
C. 30
D. 60
E. 75

Important property:

A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s side, then that triangle is a right triangle. For more check Triangles and Circles chapters of Math Book: math-triangles-87197.html, math-circles-87957.html.

According to the above, since PR is the diameter of the circle (it passes through the center), then triangle PQR is a right triangle with right angle at Q.

Next, \(\angle{OQR}=\angle{PQR}-\angle{PQO}=90^{\circ}-60^{\circ}=30^{\circ}\).

Finally, since \(OQ=OR=radius\), then triangle OQR is an isosceles, so \(\angle{OQR}=\angle{ORQ}=30^{\circ}\).

Answer: C.

Hope it's clear.
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Re: O is the center of the circle. What is <PRQ?  [#permalink]

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New post 03 Sep 2012, 11:07
Since one side of the triangle passes through the center of the circle we know that it is a 30-60-90 triangle.
Q = 90, P = 60, R = 30
The correct answer is C.
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Re: O is the center of the circle. What is <PRQ?  [#permalink]

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New post 29 Sep 2014, 02:49
Attachment:
circle.png
circle.png [ 21.62 KiB | Viewed 5596 times ]


Radius of circle = OP = OQ = OR

So, \(\angle OPQ = 60\)

Let \angle PRQ = x

60 + 60 + x + x = 180

x = 30

Answer = C
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Re: O is the center of the circle. What is <PRQ?  [#permalink]

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New post 13 Nov 2019, 00:08
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Re: O is the center of the circle. What is <PRQ?   [#permalink] 13 Nov 2019, 00:08
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