Question

The lengths of the sides of a triangle are in the ratio $3:4:5$. Find the area of the triangle if its perimeter is $144$ cm.

Answer 1

Let the sides of the triangle be $3a,4a,5a$.
Perimeter of the triangle $=$ Sum of all sides $=3a+4a+5a=12a$
Given, perimeter of the triangle $=144$ cm
$\Rightarrow 12a=144$ cm
$\Rightarrow a=12$ cm

So, the sides of the triangle are $3a=36$ cm, $4a=48$ cm, $5a=60$ cm.

Area of triangle with sides $a,b,c$ and $s=\frac{a+b+c}{2}$ is $\sqrt{s(s-a)(s-b)(s-c)}$

For triangle with sides, $36,48,60$, $s=\frac{36+48+60}{2}=72$ cm.

Area of the triangle with sides 36, m, 48 cm and 60 cm $=\sqrt{72(72-36)(72-48)(72-60)}$

$=\sqrt{72\times 36\times 24\times 12}$

$=\sqrt{36\times 2\times 36\times 12\times 2\times 12}$

$=36\times 12\times 2=864{cm}^2$