Question

The diagonal of a square is twice the side of equilateral triangle the ratio of area of the triangle to the area of square is? (a) sqrt of3:8 (b) sqrt of 2:5 (c) sqrt of 3:6 (d) sqrt of 2:4

Answer 1

Let the length of the equilateral triangle be x.

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Find the area of the equilateral triangle:

Area of equilateral triangle = √3/4 a² where a is the length of the side.

Area of the equilateral triangle = √3/4 x²

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Find the area of the square:

Length of diagonal = 2x

Area = 1/2 (diagonal 1 )(diagonal 2)

Area = 1/2 (2x)²

Area = 1/2 (4x²)

Area = 2x²

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Find ratio of triangle to square:

triangle : square = √3/4 x² : 2x²

Divide both sides by x²:

triangle : square = √3/4 : 2

Multiply both sides by 4:

triangle : square = √3 : 8

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Answer: The ratio is (A) √3 : 8

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Answer 2

Let x be the length of the side of equilateral triangle

SO,

Area of equilateral triangle of side of length x is given by

=(√3 x)/4

Now,

Length of diagonal of the square = 2x

So,

Area of square = 1/2( Product of the length of diagonals)

Since in square both diagonals are of equal length ,

Area = 1/2 × (4x²)

=x²/2

So,

Ratio of the areas = (√3x²/4) / (x² /2)

=√3/8

This is the required solution