Question

Q54. The diagonal of a square is twice the side of equilateral
triangle. Find the ratio of Area of the Triangle to the Area of Square.
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Answer 1

Given that the diagonal of a square is twice the side of a equilateral triangle.

Let the side of the square be x metres.

Using Pythagoras theorem,

⇒ Diagonal = √{ (side)² + (side)² }

⇒ Diagonal = √{ (x)² + (x)² }

⇒ Diagonal = √( 2x² )

⇒ Diagonal = √2 x

Now, The diagonal of the square is twice the side of the equilateral triangle, So

⇒ 2 × Side of equilateral triangle = Diagonal

⇒ Side = Diagonal / 2

⇒ Side = √2x / 2

Now, We know

⇒ Area of Square = (Side)²

⇒ Area of Square = x² ...[1]

Similarly,

⇒ Area of Equilateral triangle = √3/4 × (Side)²

⇒ Area of triangle = √3/4 × (√2x/2)²

⇒ Area of triangle = √3 / 4 × 2x² / 4

⇒ Area of triangle = √3x² / 8 ...[2]

Hence, Ratio of the areas of the triangle to the square can be calculated as,

⇒ [2] / [1]

⇒ (√3x² / 8) / x²

⇒ √3x² / 8x²

⇒ √3 / 8

∴ Ratio of areas is √3 : 8

Answer 2

Step-by-step explanation:

Let the length of the side of the equilateral triangle be x

And The diagonal of the square will be 2x.

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Now,first find the length of square As,

finding length of square

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using Pythagoras theorem,

b²+p²=h²

Now,As per question

a² + b² = c²

Length² + Length² = (2x)²

2Length² = 4x²

Length² = 4x²/2

Length²=2x²

Now length,

Length =√2x

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Now,we will find the area of the square which is side² or side ×side.

Area=side²

Now,

Area = Length²

Area = [ (√2) x ] ²

Now, we have,

Area = 2x² units²

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Now,find the area of the equilateral triangle

As we know, Area of equilateral∆=√3/4 a²

Area of equilateral∆=√3/4 a²Now,

Area = √3/4 (Side)²

Area = √3/4x² units²

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Now, finding ratio of this number

Area of square = √3/4x² : 2x²

Divide by x² both side,you get

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Area of the triangle : Area of square = √3/4 : 2

Divide by 4 both side ,you get

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Area of the triangle : Area of square = √3 : 8

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Hence,Answer is √3:8