A square of highest possible area and an equilateral triangle of highest possible area are inscribed in a circle. what is the ratio of the area of the square to the area of a equilateral Triangle.
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Answers
Step-by-step explanation:
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)
Answer:
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