Question

If every side of a triangle is doubled, then increase in the area of the triangle is

Options

• \[100\sqrt{2} \] %

   

• 200%

•  300%

•  400%

Answer 1

The area of a triangle having sides a, b, c and s as semi-perimeter is given by,

`A = sqrt(s(s-a)(s-b)(s-c))`, where

`s = (a+b+c)/2 ⇒ 2s = a + b + c`

We take the sides of a new triangle as 2a, 2b, 2c that is twice the sides of previous one

Now, the area of a triangle having sides 2a, 2b, and 2c and  s₁ as semi-perimeter is given by,

`A_1 = sqrt(s_1(s_1-2a)(s_1 - 2b)(s_1 - 2c) `

Where,

`s_1 = (2a +2b+2c)/2`

`s_1 = (2(a+b+c))/2`

s₁ = a + b + c 

s₁ = 2s

Now,

`A_1 = sqrt(2s (2s-2a)(2s-2b)(2s-2c))`

`A
_1 = sqrt(2s xx 2 ( s-a)xx 2 ( s-b) xx 2 (s-c))`

`A_1 = 4 sqrt(s (s-a)(s-b)(s-c))`

`A_1 = 4A`

Therefore, increase in the area of the triangle

=A₁ - A

=4A - A 

=3 A Percentage increase in area 

`= (3A)/A xx 100`

= 300 %