Question

If each side of a triangle is doubled, the find percentage increase in its area.

 

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

=3A

Percentage increase in area 

`=(3A)/A xx 100 `

= 300%