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Finding length of a line inscribed in a circle (GMATPrep1) [#permalink]

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18 Sep 2009, 12:54

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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.

Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink]

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18 Sep 2009, 22:17

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Figure given is modified with some details.

Attachment:

Untitled.jpg [ 14.74 KiB | Viewed 8559 times ]

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

Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink]

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19 Sep 2009, 17:51

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.

Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink]

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21 Sep 2009, 00:40

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.

Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink]

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05 Oct 2009, 10:54

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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

Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink]

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05 Oct 2009, 15:04

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

Re: Finding length of a line inscribed in a circle (GMATPrep1) [#permalink]

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05 Oct 2009, 19:53

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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