Circles inside a circle : PS Archive
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# Circles inside a circle

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Senior Manager
Joined: 21 Jul 2009
Posts: 265
Location: New York, NY
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Kudos [?]: 105 [0], given: 23

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07 Oct 2009, 15:27
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Question Stats:

86% (01:37) correct 14% (00:00) wrong based on 3 sessions

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Three circles of radius 1 are externally tangent to each other and internally
tangent to a larger circle. What is the radius of the larger circle?
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Senior Manager
Joined: 18 Aug 2009
Posts: 328
Followers: 8

Kudos [?]: 288 [0], given: 13

Re: Circles inside a circle [#permalink]

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07 Oct 2009, 15:39

The centers of smaller internal circles create an equilateral triangle, with side S = 2.
Intersection point of medians of this triangle lie on the origin/center of the larger circle.
So the radius of the big circle = 2/3*(length of median) + (radius of the smaller circle).
$$R = 2/3*(\sqrt{3}/2)*S + 1 = (2/\sqrt{3}) + 1 = (3 + 2\sqrt{3})/3$$
Senior Manager
Joined: 21 Jul 2009
Posts: 265
Location: New York, NY
Followers: 3

Kudos [?]: 105 [0], given: 23

Re: Circles inside a circle [#permalink]

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08 Oct 2009, 08:28
I don't have an OA, but I'm getting D too.

I put an equilateral triangle with a side of 2 with vertices in the centers of the small circles. The circumcircle of an equilateral triangle is $$\frac{S}{\sqrt{3}}$$, so the distance between the vertices and the center of the large circle is $$\frac{2}{\sqrt{3}}$$. Now add the distance between the vertices to the large circle, which is 1.
$$\frac{2}{\sqrt{3}}+1$$
Which is equal to D.
Intern
Joined: 01 May 2009
Posts: 23
Followers: 0

Kudos [?]: 0 [0], given: 2

Re: Circles inside a circle [#permalink]

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12 Oct 2009, 00:19
hi, agree, i got D as well
Re: Circles inside a circle   [#permalink] 12 Oct 2009, 00:19
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