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Square in Rectangle

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Square in Rectangle

by theCodeToGMAT » Sat Sep 28, 2013 6:05 am
I don't have the OA for this question

Q. What is the side of the Square of the largest size that can be symetrically inscribed in the equilateral triangle of side 12.

My Answer : [spoiler]12√3 [ 2 - √3][/spoiler]
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by Brent@GMATPrepNow » Sat Sep 28, 2013 7:27 am
theCodeToGMAT wrote:I don't have the OA for this question

Q. What is the side of the Square of the largest size that can be symetrically inscribed in the equilateral triangle of side 12.
Here's how all of this looks.
Image

Notice that we have a special 30-60-90 right triangle hiding in the corner.
Image

Let's say the shortest side of the right triangle has length x
Image

If we compare this triangle to the "base" 30-60-90 right triangle, we can see that the other 2 sides must have lengths 2x and √3x
Image

Since all 4 sides of the square have the same length, we know that sides all have length √3x
Image

Finally, by symmetry, we know that the bottom side of 30-60-90 right triangle on the LEFT SIDE also has length x
Image

So, side BC has length x + x + √3x
We're told that each side of the equilateral triangle has length 12, so we can write:
x + x + √3x = 12
simplify: 2x + √3x = 12
factor: x(2 + √3) = 12
divide both sides by (2 + √3) to get: x = 12/(2 + √3)

IMPORTANT: Since each side of the square has length √3x, the final answer = √3[12/(2 + √3)]
= [spoiler]12√3/(2 + √3)[/spoiler]

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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