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

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Competition Mode Question https://gmatclub.com/forum/download/file.php?id=54172 A square is inscribed in an equilateral triangle as shown above. What is the ratio of the area of the square to that of the equilateral triangle? A. 4: (12 + 15√3) B. 6 : (6 + 5√3) C. 12 : (12 + 7√3) D. 18 : (12 + 15√3) E. 24 : (24 + 7√3) Are You Up For the Challenge: 700 Level Questions Untitled.png

ArunSharma12 · Answer

A square is inscribed in an equilateral triangle as shown above. What is the ratio of the area of the square to that of the equilateral triangle?

A. 4:(12+15√3)

B. 6:(6+5√3)

C. 12:(12+7√3)

D. 18:(12+15√3)

E. 24:(24+7√3)

Attachment:

img.png [ 8.03 KiB | Viewed 27660 times ]

angles of the triangle mentioned in the figure are in ratio 30:60:90 so the sides will be in the ratio x:√3x:2x.

side of equilateral triangle = 2x +√3x
side of square = √3x

Area(square)/Area(Equilateral triangle) = \(\frac{(x√3)^2*4}{(2x+√3x)^2*√3}\) = \(\frac{3x^2*4}{x^2*(7+4√3)*√3}\) = \(\frac{4√3}{(7+4√3)}=\frac{12}{(7√3+12)}\)
Ans: C

freedom128 · Answer

See attached self-explanatory sketch. Let side of the square be  a

Square area = a^2

Triangle:
base= a*(1+2/√3)
height= a*(1+ √3 /2)
area= a^2 *(1+ 7√3 /12)

Square area/ Triangle area
= a^2 / {a^2 *(1+ 7√3 /12)} 
= 1 / (1+ 7√3 /12)
= 12 / (12+ 7√3)

FINAL ANSWER IS (C)

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ConquerorGmat · Answer

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.

Posted from my mobile device

lacktutor · Answer

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

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

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