If circle O is inscribed inside of equilateral triangle T, w : GMAT Problem Solving (PS)
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# If circle O is inscribed inside of equilateral triangle T, w

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If circle O is inscribed inside of equilateral triangle T, w [#permalink]

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12 Jan 2014, 19:19
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If circle O is inscribed inside of equilateral triangle T, which of the following expresses the ratio of the radius of circle O to one of the sides of triangle T?

A) 1 to 2
B) 1 to $$\sqrt{2}$$
C) 1 to $$\sqrt{3}$$
D) 1 to 2$$\sqrt{2}$$
E) 1 to 2$$\sqrt{3}$$

No diagram is provided.
[Reveal] Spoiler: OA

Last edited by Engr2012 on 21 Sep 2015, 16:33, edited 1 time in total.
Formatted the question
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Re: If circle O is inscribed inside of equilateral triangle T, w [#permalink]

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13 Jan 2014, 09:54
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se7en14 wrote:
If circle O is inscribed inside of equilateral triangle T, which of the following expresses the ratio of the radius of circle O to one of the sides of triangle T?

1 to 2
1 to $$\sqrt{2}$$
1 to $$\sqrt{3}$$
1 to $$2\sqrt{2}$$
1 to $$2\sqrt{3}$$

No diagram is provided.

Dear se7en14,
I'm happy to help.

Here's a diagram:
Attachment:

equilateral with inscribed circle.JPG [ 15.35 KiB | Viewed 1507 times ]

Point E is the center of the circle, so DE is the radius. Let's say that DE = 1. Notice that triangle DEC is a 30-60-90 triangle, with a 30 degree angle at C and a 60 degree angle at E. For more on the properties of this triangle, see:
http://magoosh.com/gmat/2012/the-gmats- ... triangles/
The sides have ratios of 1-2-sqrt(3). Here:
DE = 1
CE = 2
DC = $$\sqrt{3}$$
Now, notice that DC is half the side, because D is a midpoint of the side. This means
AC = 2*(DC) = $$2*\sqrt{3}$$
That's the length of the side. Therefore,
radius:side = 1: $$2*\sqrt{3}$$

Does all this make sense?
Mike
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Mike McGarry
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Re: If circle O is inscribed inside of equilateral triangle T, w [#permalink]

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13 Jan 2014, 10:10
@Mike

Yes, thanks!
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21 Sep 2015, 16:02
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Re: If circle O is inscribed inside of equilateral triangle T, w [#permalink]

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21 Sep 2016, 20:24
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If circle O is inscribed inside of equilateral triangle T, w [#permalink]

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22 Sep 2016, 01:59
se7en14 wrote:
If circle O is inscribed inside of equilateral triangle T, which of the following expresses the ratio of the radius of circle O to one of the sides of triangle T?

A) 1 to 2
B) 1 to $$\sqrt{2}$$
C) 1 to $$\sqrt{3}$$
D) 1 to 2$$\sqrt{2}$$
E) 1 to 2$$\sqrt{3}$$

No diagram is provided.

Check out this post: http://www.veritasprep.com/blog/2013/07 ... other-way/

It discusses the relation between radius and side when a circle is inscribed in a regular polygon such as equilateral triangle, square etc.
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Re: If circle O is inscribed inside of equilateral triangle T, w   [#permalink] 22 Sep 2016, 01:59
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