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# Another circle problem

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Manager
Joined: 21 Jul 2009
Posts: 242

Kudos [?]: 141 [0], given: 23

Location: New York, NY
Another circle problem [#permalink]

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08 Oct 2009, 09:14
00:00

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0% (00:00) correct 100% (00:42) wrong based on 1 sessions

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Circles A, B, and C are externally tangent to each other and internally
tangent to circle D. Circles B and C are congruent. Circle A has radius 1
and passes through the center of D. What is the radius of circle B?
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Kudos [?]: 141 [0], given: 23

Math Expert
Joined: 02 Sep 2009
Posts: 42620

Kudos [?]: 135695 [0], given: 12706

Re: Another circle problem [#permalink]

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08 Oct 2009, 12:39
Were did you get this one?

Well, you can guess how to solve this problem but with calculation it will take way more time than 2 min.

Here is solution:

Let A, B and C be the centers of the circles A,B,C.
O the center of the biggest circle. P intersection of B and C.
R the radius of biggest circle=1+1=2, r radius of B=radius of C.

--> Triangles ABP and OBP are right angle.

In right OBP OB=R-r=2-r=(hypotenuses), BP=r OP=x --> (2-r)^2=r^2+x^2 --> x=(4-4r)^1/2

In right ABP AB=1+r=(hypotenuses), BP=r, AP=1+x=1+(4-4r)^1/2 --> (1+r)^2=(1+(4-4r)^1/2)^2+r^2 -->
r(9r-8)=0 --> r=8/9

Answer D.

I really doubt that there is a faster solution.
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Kudos [?]: 135695 [0], given: 12706

Manager
Joined: 21 Jul 2009
Posts: 242

Kudos [?]: 141 [1], given: 23

Location: New York, NY
Re: Another circle problem [#permalink]

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08 Oct 2009, 13:31
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Bunuel wrote:

In right OBP OB=R-r=2-r=(hypotenuses),

I got stuck at this point... don't understand this.

Question is from a geometry quiz, not necessarily GMAT. I don't have an OA either (for those who expect one).

Kudos [?]: 141 [1], given: 23

Math Expert
Joined: 02 Sep 2009
Posts: 42620

Kudos [?]: 135695 [0], given: 12706

Re: Another circle problem [#permalink]

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08 Oct 2009, 14:00
mendelay wrote:
Bunuel wrote:

In right OBP OB=R-r=2-r=(hypotenuses),

I got stuck at this point... don't understand this.

Question is from a geometry quiz, not necessarily GMAT. I don't have an OA either (for those who expect one).

OK, OB is the distance from center of biggest circle to the center of circle B. If we continue the line OB we'll get to the tangent point of biggest circle and circle B, call it O1. OO1 is a radius of the biggest circle and BO1 is a radius of the circle B, so OB=OO1-BO1=2-r.

Hope now it's clear. Well as I said the problem I see in this question is calculation part, because if you draw it correctly with all points, it won't be as hard as it seems.

As for OA. The way of solving is correct, check the calculation and if your answers match mine than it's the one.
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Kudos [?]: 135695 [0], given: 12706

Re: Another circle problem   [#permalink] 08 Oct 2009, 14:00
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# Another circle problem

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