What is the radius of the circle? : GMAT Problem Solving (PS)
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# What is the radius of the circle?

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09 Jul 2011, 22:35
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Question Stats:

67% (00:00) correct 33% (00:00) wrong based on 3 sessions

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In the figure above, three circles of radius "r" are tangent to one another. The area of the shaded region is 100\sqrt{3} - 50pie . What is the value of "r"?

A. 5
B. 10
C. 20
D. 25
E. 50
[Reveal] Spoiler: OA

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Vijendra Lakshmana Rao,
GMAT score target is 700

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09 Jul 2011, 23:06
Not able to see the image.. can you repost it?
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09 Jul 2011, 23:37
I have attached the image.
Attachments

DYGMATQR2p81_145A.GIF [ 3.05 KiB | Viewed 2605 times ]

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Regards,
Vijendra Lakshmana Rao,
GMAT score target is 700

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10 Jul 2011, 02:49
Area of shaded region = area of the triangle - area of 3 sectors
the triangle is an equilateral triangle with side 2r.thus the central angle is 60 each

area of each sector is 60/360 * area of circle = 1/6 (area of cirle) = Pi.r^2/6

area of 3 sectors= pi.r^2/2

area of triangle = r^2 (sqrt{3})/4

shaded area is r^2 (sqrt{3})/4 - pi.r^2/2

plugging in values for r, for r=10 area - 100(sqrt 3) - 50 Pi
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10 Jul 2011, 03:22
2
KUDOS
Ok - Here is how I did this question. I have put the alphabets in the figure above:

So lets say o --> Centre of the circle
P--> Left hand side where two circle meets
A--> Left hand corner of the Triangle
M--> Where the bottom two circle meets
Q--> Right Hand Side where two circle meets

To answer this question one has to be look at the diagram closely and should be familiar with the area of sector formula, area of circle formula and area of triangle formula.

So if you see my naming conventions there are three sectors in the figure which are:

OPQ, APM and QBM.

From the figure its clear that Area of Shaded area = Area of triangle - Area of 3 sectors which are named above.

Now we have to figure out what kind of triangle is this.

So, OA=AP+OP.
and AP=OP {because the radius of circles are always equal). let say r be the radius. So AP=OP=r and OA = 2r

Same applies to other two sides of the Triangle. Therefore, we can conclude that the Triangle is an equivalent triangle and the sum of all the three angles is 180 degrees and each angle is of 60 degree in that case. Therefore angle O is of 60 degrees.

Area of sector = Pi.r^2(angle)/360 = pi (r)^2 * 60/360 {Substituuing the values of r and angle which are 2r and 60} ==> pi r^2 /6

Area of one sector => pi r^2 /6
Area of three sectors ==>pi r^2/2

Area of Equilateral triangle {Triangle is equilateral because every angle is 60 degrees and sides corresponding to equal angles are also equal)

Therefore area of equilateral triangle = Side^2 * Square root3/4

Area of shaded region = Side^2 * Square root3/4 minus pi r^2/2.
Plug in the all values of above to give you area 100\sqrt{3} - 50pi. when r=10 you will get this value
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Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730

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10 Jul 2011, 04:05
Amazing explanation enigma123.... Kiddos!!!
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Regards,
Vijendra Lakshmana Rao,
GMAT score target is 700

Re: What is the radius of the circle?   [#permalink] 10 Jul 2011, 04:05
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