Question

If the length of each side of an equilateral triangle is increased by 1 meter then its area will be increased by √3sq.m. Let us write by calculating the length of side of equilateral triangle.​

Answer 1

Answer :

3 / 2 m

Explanation :

Let the side of an equilateral triangle be 'a' m

Area of an equilateral triangle = √3 a²/4 sq.m

If the length of each side us increased by 1 m

Side of the newly formed equilateral triangle = ( a + 1 ) m

Area the newly formed equilateral triangle = √3 ( a + 1 )² / 4 m²

Given :

Area of the newly formed equilateral is increased by √3 m²

⇒ √3 ( a + 1 )² /4 = √3 a²/4 + √3

⇒ √3 ( a + 1 )² /4 = √3 ( a²/4 + 1 )

⇒ ( a² + 2a + 1 ) / 4 = ( a² + 4 ) / 4

⇒ a² + 2a + 1 = a² + 4

⇒ 2a + 1 = 4

⇒ 2a = 3

⇒ a = 3 / 2

Therefore the length of the side of the equilateral triangle is 3 / 2 m.

Answer 2

Given :-

• Length of each side increased by 1 m

• Area of equilateral∆ increased by √3 m²

To find :-

• Length of side of equilateral∆ = ?

Solution :-

Let the side of equilateral∆ be x m

As we know,

☛ $\boxed{\boxed{\sf\blue{Area \: of \: equilateral\: \triangle =\dfrac{\sqrt{3}}{4}\: a^2 }}}$

Here, a = length of each equal sides.

Case l : —

→ Area of equilateral∆ = √3/4 (x²)

Case 2 : —

As length will be increased by 1 m

Area will be increased by √3 m²

A/q,

→ √3/4 (x + 1)² = √3/4 (x²) + √3

→ [√3(x + 1)²]/4 = (√3x²)/4 + √3

→ [√3 (x² + 2x + 1)]/4 = [√3x² + 4√3]/4

→ [√3x² + 2x√3 + √3]/4 = [√3x² + 4√3]/4

Cancelling out 4 from both sides —

→ √3x² + 2x√3 + √3 = √3x² + 4√3

Cancelling out √3x² from both sides:-

→ 2x√3 + √3 = 4√3

→ 2x√3 = 4√3 - √3

→ 2x√3 = 3√3

→ 2x = 3√3/√3

→ 2x = 3

→ x = 3/2

Therefore,

Side of equilateral∆ is (3/2) m.