Question

The lengths of the sides of a triangle are in the ratio $3:4:5$ and its perimeter is $144cm$. Find the area of the triangle and the height corresponding to the longest side.

Answer 1

Given: Ratio of sides of a triangle $=3:4:5$

Perimeter $=144cm$

Let the sides of a given triangle be $a=3x,$
$b=4x,c=5x$

Given: Perimeter of triangle $=144cm$

$2s=a+b+c$

$a+b+c=144$

$3x+4x+5x=144$

$12x=144$

$x=12$

So, the sides of a triangle are

$a=3\times 12=36cm$

$b=4\times 12=48cm$

$c=5\times 12=60cm$

$s=\frac{\text{Perimeter}}{2}=\frac{144}{2}=72cm$

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

$=\sqrt{72(72-36)(72-48)(72-60)}$

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

$=\sqrt{12\times 3\times 2\times 12\times 3\times 12\times 2\times 12}$

$=12\times 12\times 3\times 2$

$=864c{m}^2$

Let the length of altitude on the longest side corresponds to $60cm$ is $hcm$

We know that,

Area $=\frac{1}{2}\times \text{base}\times \text{height}$

$\Rightarrow \text{Height}\left(h\right)=\frac{area\times 2}{base}=\frac{864\times 2}{60}=\frac{1728}{60}=28.8cm$

Hence, the length of altitude is $28.8cm$.