A square is inscribed in an equilateral triangle as shown above. What

Answer-C
Use tan60 to get side of triangle if you are assuming square side to be X.
So side of equi. triangle is
X + 2*x/tan60.
Now ratio of area=area of square (side x)/area of triangle (X+2*X/tan60).after calculation and rationalisation..we will get Ans.c.

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∆ ABC is an equilateral triangle
Let (x+y) be the side of the triangle.
Let x be the side of the square.

∆ ABC and ∆ DBE are the similar triangles.
--> ∆ DBE an equilateral triangle too. (x is the side of that triangle)

Well, ∆ BHE and ∆ EFC are right-angled triangles and they are similar:

\(\frac{EF}{BH}= \frac{EC}{BE}\)

---> \(\frac{x}{(√3/2)x}= \frac{y}{x}\)

\(\frac{y}{x} =\frac{2}{√3}\)
--> \(BC=x+y = x+ (\frac{2}{√3})x= x( \frac{2+√3}{√3})\)

So, The ratio of the area of the square to that of the triangle is
\(\frac{x^{2}}{ (√3/4)(y^{2})}= \frac{x^{2}}{ (√3/4)(x^{2}(2+√3)^{2}/(√3)^{2}}= \frac{4√3}{(2+√3)^{2}}= \frac{4√3}{(7+4√3)}= \frac{12}{ 12+7√3}\)

Answer (C).

PLEASE SEE THE ATTACHED FILE.

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