Question

If height of an equilateral triangle is doubled, what percent of area will be
increased of this triangle?​

Answer 1

Answer: 100%

An Equilateral triangle is a triangle that has its all three sides are equal in length.

Let ∆ABC be a triangle that has its all three sides equal to a.

Now,

⇒ Area of an equilateral triangle

• = √3a² /4 ...(1)

Now, Let another triangle that has its height doubled as compared to the first triangle. i.e., 2√3a /2 = √3a

But, Here it is worth thinking that the side of the new triangle wouldn't be a. So, Let the side of the new triangle be h.

Using Pythagoras theorem,

⇒ Hypotenuse² = Height² + Base²

⇒ h² = (√3a)² + (h/2)²

⇒ h² - h²/4 = 3a²

⇒ 3h² / 4 = 3a²

⇒ h² / 4 = a²

⇒ h² = 4a²

⇒ h = 2a

Because, Distance can't be negative, Hence h = -2a is neglected.

Now,

⇒ Area of triangle = 1/2 × Base × Height

⇒ Area = 1/2 × Base × √3a

Because, The height bisects the base when dropped from the top vertex.

Therefore, Base = side/2

The base of the new triangle is h/2 . So, In terms of a, Base = a

⇒ Area = 1/2 × a × √3a

⇒ Area = √3a² / 2 ...(2)

Now, % Increase in area

⇒ (2) / (1) × 100

⇒ (√3a²/2) / (√3a² / 4) × 100

⇒ 4√3a² / 2√3a² × 100

⇒ 4/2 × 100

⇒ 200 %

Which means, a 100% of area will be increased. The area will also be doubled.

Hence, The percentage increase in area will be 100%.

Answer 2

Given :-

Height of equilateral triangle is doubled

To Find :-

Increase in area

Solution :-

Let the height be h

Area = √3a²/4

Now

When height doubled

Area = 2√3a/2 = √3a

Now

Using Pythagoras theorem

h² = (√3a)² + (h/2)²

h² = 3a + h²/4

h² - h²/4 = 3a²

4h² - h²/4 = 3a²

3h²/4 = 3a²

h² = 4a²

h = √4a²

h = 2a

Area =  1/2 × Base × √3a

Base = Side/2

Area (new) =  1/2 × a × √3a

= √3a² / 2

Now

Increased percentage

(√3a²/2) / (√3a² / 4) × 100

4√3a² / 2√3a² × 100

• Cancelling √3a²  

4/2 × 100

4 × 25

100%