Question

The sides of a triangle are in the ratio $1:2:2$ and its perimeter is $500\text{m}$. Then, area of the triangle is:

• A. 2500$\sqrt{15}{m}^2$
• B. 100$\sqrt{15}{m}^2$
• C. 800$\sqrt{15}{m}^2$
• D. 200$\sqrt{15}{m}^2$

Answer 1

The correct option is A

2500$\sqrt{15}{m}^2$

Given, the ratio of sides is $1:2:2$.
Let the common factor be ‘x’
Then, $1x+2x+2x=500$
$\Rightarrow 5x=500$
$\Rightarrow x=100$ m
$\therefore \text{The lengths of the sides are 100 m, 200 m and 200 m.}$
Then,
$s=\frac{(100+200+200)}{2}$ = 250 m

The area (A) of a triangle can be calculated using Heron's formula, given by:
$A=\sqrt{s(s-a)(s-b)(s-c)}$,
where a, b and c are its sides and s is its semi-perimeter.

$A=\sqrt{250(250-100)(250-200)(250-200)}=\sqrt{250\left(150\right)\left(50\right)\left(50\right)}=\sqrt{\left(25\right)\left(15\right)\left(25\right)\left(10000\right)}=2500\sqrt{15}{\text{m}}^2$
Therefore, the area of the required triangle is $2500\sqrt{15}{\text{m}}^2$.