Question

The sides of a triangle are in the ratio $3:5:7$ and its perimeter is $300m.$ find the area of the triangle.

Answer 1

Given: Ratio $=3:5:7$

Let ${}^{\prime }x;$ be the constant ratio.

Then, the sides of the given triangle are $3x,5x,$ and $7x.$

Perimeter $=300m$ (Given)

$3x+5x+7x=300$ [$\because$ Perimeter $=$ Sum of all sides ]

$\Rightarrow 15x=300$

$\Rightarrow x=\frac{300}{15}=20$

Sides are:

$3x=3\left(20\right)=60m$
$5x=5\left(20\right)=100m$
$7x=7\left(20\right)=140m$
Perimeter $=300m$
Semi-perimeter, $s=\frac{300}{2}=150m$

Here, the three sides are

$a=60m,b=100m,c=140m$

or, Semi-perimeter, $s=\frac{60+100+140}{2}=\frac{(60+100+140)}{2}=150$
Using Heron's Formula, Area of the triangle $=\sqrt{s(s-a)(s-b)(s-c)}$

$=\sqrt{150(150-60)(150-100)(150-140)}$

$=\sqrt{150\left(90\right)\left(50\right)\left(10\right)}$

$=\sqrt{15\times 15\times 3\times 100\times 100}$

$=1500\sqrt{3}$

$\therefore$ Area of the given triangle $=1500\sqrt{3}{m}^2$