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: \(2*\sqrt{3}\)

Answer =

(D)Does all this make sense?

Mike

_________________

Mike McGarry

Magoosh Test Prep