Question

A square is inscribed in an equilateral triangle. What is the ratio of
the area of the square to that of the equilateral triangle?
A) 12 : 12 + 7√3
B) 24 : 24 + 7√3
C) 18 : 12 + 15√3
D) 6 : 6 + 5√3

solve with step by step explanation and formula used

Answer 1

GIVEN: An equilateral triangle ABC.

Let A square DEFG, each side = ‘a’ unit is inscribed in the triangle, covering maximum area, EF is perpendicular to BC. AH is perpendicular to DE.

DE ll BC, So, angle AED = angle ACB = 60°

In triangle EFC

Sin60° = a/ EC

=> √3/2 = a/EC

=> EC = 2a/√3………….(1)

In triangle AHE

Cos 60° = a/2 / AE

=> 1/2 = a/2 /AE

=> AE = a……………(2)

So, side AC of the triangle = 2a/√3 + a

=> AC = (2+√3)a /√3

=> AC = (2√3 + 3)a /3 ………..(3)

Now area (squareDEFF) = a² ……….(4)

And area ( equilateral triangle ABC) = √3side² /4

=> √3 * AC² * 1/4

=> √3* (2√3 + 3)²a² /9 * 1/4

=> √3 * ( 12 + 9 + 12√3) a² / 9 * 1/4

=> ( 21√3 + 36) a²/9 * 1/4

=> area( triangleABC) = (21√3 + 36)a² / 36………(5)

Now area(Square) : area(eq.triangle) = eq(4) /eq(5)

=> a² / [( 21√3 + 36) a²/36]

=> 36a² / ( 21√3 + 36)a²

=> 12 / 7√3+12

12 : (7√3+12)