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# In the circle above, PQ is parallel to diameter OR, and OR

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Manager
Joined: 24 Jun 2009
Posts: 60
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Kudos [?]: 23 [0], given: 2

In the circle above, PQ is parallel to diameter OR, and OR [#permalink]  27 Nov 2009, 11:12
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Difficulty:

65% (hard)

Question Stats:

69% (01:51) correct 31% (01:47) wrong based on 16 sessions
I got this right but I eyeballed it. Does anyone know the proper way to solve this problem?

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-the-circle-above-pq-is-parallel-to-diameter-or-93977.html
[Reveal] Spoiler: OA
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Manager
Joined: 29 Oct 2009
Posts: 211
GMAT 1: 750 Q50 V42
Followers: 78

Kudos [?]: 791 [0], given: 18

Re: Tough GMATPrep geo question - length of minor arc PQ [#permalink]  27 Nov 2009, 11:43
Assume centre of the circle to be point C.

Since angle ORP = $$35^{\circ}$$ , angle OCP will be = $$70^{\circ}$$

Note: This follows from the circle property that an inscribed angle (angle ORP in our case) is exactly half the corresponding central angle (angle OCP in our case).

Now, we can calculate the length of arc OP :

Length of arc OP = (angle OCP)*(radius of circle) = $$\frac{7\pi}{18}*9$$ = $$\frac{7\pi}{2}$$
Note: An angle of $$70^{\circ}$$ corresponds to $$\frac{7\pi}{18}$$ radians.

Also, since chord PQ is parallel to the diameter, arc OP must be equal to arc QR.

Therefore, total length of arc OP + QR = $$\frac{7\pi}{2}+\frac{7\pi}{2}$$ = $$7\pi$$

Now, length of arc PQ will be = half the circumference of the circle - combined length of arcs OP and QR

Half the circumference of the circle = $$\pi*r$$ = $$9\pi$$

Thus length of arc PQ = $$9\pi - 7\pi$$ = $$2\pi$$

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VP
Joined: 05 Mar 2008
Posts: 1473
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Kudos [?]: 216 [0], given: 31

Re: Tough GMATPrep geo question - length of minor arc PQ [#permalink]  27 Nov 2009, 11:50
shanewyatt wrote:
I got this right but I eyeballed it. Does anyone know the proper way to solve this problem?

OCP will equal 70 so qcr = 70 and you know the part under the diameter = 180

140+180 = 320

PQ = 360-320/360

40/360 = 1/9
circumference = 18 pi
1/9*18pi = 2pi
Re: Tough GMATPrep geo question - length of minor arc PQ   [#permalink] 27 Nov 2009, 11:50
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