Question

The sides of a triangle are in the ratio $1:2:2$ and its perimeter is $500m$. The area of the triangle in $m^2$ is:

• A. 500$\sqrt{\left(15\right)}$
• B. 100$\sqrt{\left(15\right)}$
• C. 800$\sqrt{\left(15\right)}$
• D. 200$\sqrt{\left(15\right)}$

Answer 1

The correct option is A

500$\sqrt{\left(15\right)}$

The area of a triangle is given by the Heron's formula, $A=\sqrt{s(s-a)(s-b)(s-c)}$
$\text{Let side a be x, then b = 2x and c = 2x}$
$\text{Perimeter, p = x + 2x + 2x = 5x = 500m}$
$\text{Thus, x = 100m}$
$a=x=100m,b=c=2x=200m$
$s=\frac{a+b+c}{2}$ = $\frac{100+200+200}{2}$
= $\frac{500}{2}=250m$
$A=\sqrt{s(s-a)(s-b)(s-c)}$
$=\sqrt{250(250-100)(250-200)(250-200)}$
$=\sqrt{250\left(150\right)\left(50\right)\left(50\right)}$
$=\sqrt{\left(25\right)\left(15\right)\left(10000\right)}$ = $500\sqrt{15}{m}^2$