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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]  12 Jan 2014, 19:19
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Difficulty:

45% (medium)

Question Stats:

70% (03:43) correct 30% (01:59) wrong based on 27 sessions
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{sq rt of 3}
1 to 2\sqrt{2}
1 to 2\sqrt{3}

No diagram is provided.
[Reveal] Spoiler: OA
Magoosh GMAT Instructor
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Re: If circle O is inscribed inside of equilateral triangle T, w [#permalink]  13 Jan 2014, 09:54
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Expert's post
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 559 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
_________________

Mike McGarry
Magoosh Test Prep

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Re: If circle O is inscribed inside of equilateral triangle T, w [#permalink]  13 Jan 2014, 10:10
@Mike

Yes, thanks!
Re: If circle O is inscribed inside of equilateral triangle T, w   [#permalink] 13 Jan 2014, 10:10
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