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Geometry

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Geometry

by tdkk123 » Sun Sep 18, 2011 9:28 pm
Need a detailed solution for the below problem

What is the side of the square of the largest size that can be symmetrically inscribed in
an equilateral triangle of side 12?

OA [spoiler]12√3 / (2 + √3)[/spoiler]
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by cans » Sun Sep 18, 2011 10:06 pm
what do you mean by symmetrically inscribed?
Because if a square is inscribed i think it means that all the four corners lie on triangle. and there are only 3 sides.
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by tdkk123 » Sun Sep 18, 2011 10:33 pm
Image


I guess it would look something like this
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by cans » Sun Sep 18, 2011 11:15 pm
but that's not symmetrical.
(Y and Z have different view)
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by cans » Sun Sep 18, 2011 11:19 pm
tdkk123 wrote:Image


I guess it would look something like this
If you take this figure,
let side of square=x. then 12 = x + 2XA (XA=DZ)
also tan 60 = x/XA -> XA = x/root(3)
12 = x + 2x/(root(3))
x = [12*root(3)] / (2 + root(3))
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by GMATGuruNY » Mon Sep 19, 2011 4:33 am
tdkk123 wrote:Need a detailed solution for the below problem

What is the side of the square of the largest size that can be symmetrically inscribed in
an equilateral triangle of side 12?

OA [spoiler]12√3 / (2 + √3)[/spoiler]
Image

Since the sides of the square in the figure above must be equal:
x√3 = 12-2x
2x + x√3 = 12
(2+√3)x = 12
x = 12/(2+√3).

Thus, each side of the square = x√3 = 12√3/(2+√3).
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