Question

The sides of a triangle are in the ratio $5:12:13$ and its perimeter is $150cm$. The area of the triangle is

(a) $375c{m}^2$

(b) $750c{m}^2$

(c) $250c{m}^2$

(d) $500c{m}^2$

Answer 1

Given:

The ratio of the sides of triangle $=5:12:13$

$\Rightarrow$ The sides of the triangle are $5x,12x$ and $13x$.

Perimeter of thr triangle $=150cm$

$\Rightarrow 5x,+12x+13x=150cm$

$\Rightarrow 30x=150cm$

$\Rightarrow x=\frac{150cm}{30}$

$\Rightarrow x=5cm$

Therefore, the sides are

$a=5\times 5cm=25cm$

$b=12\times 5cm=60cm$

$c=13\times 5cm=65cm$

Perimeter $2s=150cm$

$\Rightarrow s=\frac{150cm}{2}=75cm$

$\Rightarrow s-a=75cm-25cm=50cm$

$\Rightarrow s-b=75cm-60cm=15cm$

$\Rightarrow s-c=75cm-65cm=10cm$

By Heron's formula,

Area of triangle, $\Delta =\sqrt{s(s-a)(s-b)(s-c)}$

$\Rightarrow \Delta =\sqrt{\left(75cm\right)\times \left(50cm\right)\times \left(15cm\right)\times \left(10cm\right)}$

$\Rightarrow \Delta =\sqrt{75\times 50\times 15\times 10}c{m}^2$

$\Rightarrow \Delta =\sqrt{25\times 3\times 25\times 2\times 5\times 3\times 5\times 2}c{m}^2$

$\Rightarrow \Delta =25\times 5\times 3\times 2c{m}^2$

$\Rightarrow \Delta =750c{m}^2$

Hence, Option B is correct.