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Finding length of a line inscribed in a circle (GMATPrep1)

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Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 18 Sep 2009, 11:54
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Hello, I am completely baffled by this question from the GMATPrep1 exam. It is the only one that I still can't figure out and was seeing if anyone could shed some light on how to approach it. Basically a line segment has its two endpoints on a circle (see diagram). It is parallel to the diameter, and the diameter is 18. There is a diagonal drawn between the two lines and the diagonal forms a 35 degree angle with the diameter. From there you have to figure out what the length of the line segment is. The answer is 2∏. Thanks for any help.
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Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 18 Sep 2009, 21:17
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Figure given is modified with some details.
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From the figure,
C is the center
CN is perpendicular to PQ and hence will bisect PQ.
CP is equal to CR because both are radii and hence CRP = CPR = 35 degrees
Since PQ is parallel to CR => RPQ = PRC = 35 degrees

Now its straightforward.
Cosine NPC = Cosine 70 degrees = PN/PC = PN/9
PQ = 2*PN = 18*Cosine(70degrees)

oops our answer is in terms of PI which I dont think is correct. Please get the correct answer
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Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 18 Sep 2009, 22:08
The correct answer is 2∏. Thanks.
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Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 19 Sep 2009, 16:51
iheartiheartmath wrote:
The correct answer is 2∏. Thanks.


There should be choices if the answer is of PS type. I have just proved that the it is not equal to 2PI although it is close. here must be easier ways to eliminate the choices which one can do if knows what the choices were.
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Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 20 Sep 2009, 23:40
hi iheartiheartmath, is there an explanation to the answer '2Pi' in the GMATPrep?? I am sure the explanation must provide an approach to solve this with out using trignometry.
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Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 22 Sep 2009, 08:54
I don't think trignometry is supposed to be used to solve this problem. must be an easier way
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Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 22 Sep 2009, 09:30
GMATPrep didn't provide an explanation, just the answer, sorry.
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Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 05 Oct 2009, 09:54
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It is easy.
arcPO=arcQR=anglePEO*2/360*circumference=35*2/360*18pi=70/360*18pi=70/20pi
arcOR=Half of circumference=1/2*18pi=9pi
arcPQ=arcOR-2*arcPO=9pi-70/20pi*2=9pi-7pi=2pi
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Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 05 Oct 2009, 14:04
good one. +1 for ya.

apply4good wrote:
It is easy.
arcPO=arcQR=anglePEO*2/360*circumference=35*2/360*18pi=70/360*18pi=70/20pi
arcOR=Half of circumference=1/2*18pi=9pi
arcPQ=arcOR-2*arcPO=9pi-70/20pi*2=9pi-7pi=2pi
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Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 05 Oct 2009, 18:53
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apply4good wrote:
It is easy.
arcPO=arcQR=anglePEO*2/360*circumference=35*2/360*18pi=70/360*18pi=70/20pi
arcOR=Half of circumference=1/2*18pi=9pi
arcPQ=arcOR-2*arcPO=9pi-70/20pi*2=9pi-7pi=2pi


The original question asks for length of line segment PQ and not the arc. I guess the question meant the opposite.

There is an easier way to find the length of an arc i.e. (theta/360)*2*Pi*r where theta is the angle the arc forms at the center. In this case segment PCQ where c is the center of the circle subtends a angle of 40deg at the center of the circle. Applying it to the above formula, length of the arc is (40/360)*2*pi*9 = 2pi

Calculation of angle PCQ

PC=CR hence CPR=CRP=35deg
PQ ll OR hence QPR=PRO=35 deg
CP = CQ hence CPQ(CPR+RPQ) = PQC=70deg
PEQ = 180-CPQ-PQC = 40deg
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Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 04 Sep 2011, 10:49
Actually the answer for the length of PQ = 6.14

Angle CPN = 70Deg (35deg for isocelese Triangle + 35 deg adjacent for parallel lines)
Therefore angle PCN = 20deg (180-(90+70))

Sin 20 = PN/PC (Opp over Hyp)

PC*Sin20=PN and PC = 9 (Radius)
PN=9*Sin20
PN=3.07
PQ = 2*PN (Vertical dissects PQ)
Therefore PQ=6.14
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Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink] New post 23 Nov 2011, 08:32
Re: Finding length of a line inscribed in a circle (GMATPrep1)   [#permalink] 23 Nov 2011, 08:32
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