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?

Answer 1

Given :-

• The diagonal of a square is twice the side of a equilateral triangle.

To Find :-

• What's the ratio of area of the triangle to the area of square ?

Solution :-

• Let the side of the square be x metres.

$\qquad$ ❄ It's given, the diagonal of the square is twice the side of the equilateral triangle.

$\purple{\qquad\quad\tt ↠ 2 × Side _{ Equilateral \:triangle} = Diagonal}$

$\qquad\quad\tt ↠ Side = \dfrac{Diagonal }{ 2}$

$\qquad\quad\tt ↠ Side = \dfrac{√2x}{ 2}$

Using Pythagoras theorem:-

$\qquad\quad\tt ↠ Diagonal = \sqrt{{\bigg (side\bigg)² + \bigg(side\bigg)² }}$

$\qquad\quad\tt ↠ Diagonal = \sqrt{{ x² + x² }}$

$\qquad\quad\tt ↠ Diagonal = \sqrt{\bigg( 2x² \bigg)}$

$\qquad\quad\tt ↠Diagonal = √2 x$

We know:-

$\pink{\qquad\quad\bf ↠ Area _{Square} =\bigg (Side\bigg)²}$

$\qquad\quad\tt ↠ Area _{Square} = x² . . . . . (1)$

According to the question :-

$\pink{\qquad\quad\bf ↠ Area_{ Equilateral\: triangle }= \dfrac{√3}{4 }\times (Side)²}$

$\qquad\quad\tt ↠ Area _{ Triangle }= \dfrac{√3}{4 }\times \bigg(\dfrac{√2x}{2}\bigg)²$

$\qquad\quad\tt ↠ Area _{ Triangle } = \dfrac{√3 }{ 4} \times 2x² / 4$

$\qquad\quad\tt ↠ Area _{ Triangle } = \dfrac{√3x²}{ 8 } . . . . . (2)$

Ratio of the areas of the triangle to the square :-

$\purple{\qquad\quad\tt ↠ \dfrac{ 2} { 1}}$

$\qquad\quad\tt ↠ \dfrac{ \bigg(\dfrac{√3x² }{8}\bigg)}{ x²}$

$\qquad\quad\tt ↠ \dfrac{√3x² }{ 8x²}$

$\purple{\qquad\quad\tt ↠ \dfrac{√3 }{ 8}}$

• Ratio of areas is √3 : 8

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Answer 2

Given :

• The diagonal of a square is twice the side of equilateral triangle.

To Find :

• Find the ratio of area of the triangle to the area of square ?

Solution :

• Let the side of the square be x metres.

As we know that,

Pythagoras Theorem :

• Diagonal = √{(side)² + (side)²}

According to the Given Question :

➟ Diagonal = √{(side)² + (side)²}

➟ Diagonal = √{(x)² + (x)²}

➟ Diagonal = √(2x²)

➟ Diagonal = √2x

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

➟ 2 × Side of equilateral triangle = Diagonal

➟ Side = Diagonal/2

➟ Side = √2x/2

Now, We know

➟ Area of Square = (Side)²

➟ Area of Square = x² (Eqⁿ 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 (Eqⁿ 2)

Therefore,

• Ratio of the areas of the triangle to the square can be calculated,

➟ (2)/(1)

➟ (√3x²/8)/x²

➟ √3x²/8x²

➟ √3/8

Hence,

• the ratio of area of the triangle to the area of square √3 : 8.