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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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New post 12 Jan 2014, 20:19
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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, 17:33, edited 1 time in total.
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Re: If circle O is inscribed inside of equilateral triangle T, w [#permalink]

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New post 13 Jan 2014, 10:54
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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
equilateral with inscribed circle.JPG [ 15.35 KiB | Viewed 1055 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: \(2*\sqrt{3}\)
Answer = (D)

Does all this make sense?
Mike :-)
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Mike McGarry
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New post 13 Jan 2014, 11:10
@Mike

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

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New post 21 Sep 2015, 17:02
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Re: If circle O is inscribed inside of equilateral triangle T, w   [#permalink] 21 Sep 2015, 17:02
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