Question

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

• A. $1500m^2$
• B. $1500\sqrt{3}{m}^2$
• C. $1400m^2$
• D. $1400\sqrt{3}{m}^2$

Answer 1

Answer: (B)

The sides of a the triangular plot are in the ratio $3:5:7$.

So, let the sides of the triangle be $3x$, $5x$ and $7x$. where $x$ be the common multiplier.

Also it is given that the perimeter of the triangle is $300m$
Therefore, $3x+5x+7x=300m$

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

$\therefore x=20m$

Therefore, the sides of the triangle are: $3x=3\left(20m\right)=60m$,
$5x=5\left(20m\right)=100m$ and

$7x=7\left(20m\right)=140m$.

Now using herons formula:

Area of the triangle,

$\Delta =\sqrt{s(s-a)(s-b)(s-c)}$

Where $s=\frac{60m+100m+140m}{2}=\frac{300m}{2}=150m$

Thus, area of the triangle is:

$\Delta =\sqrt{s(s-a)(s-b)(s-c)}$

$=\sqrt{150m(150m-60m)(150m-100m)(150m-140m)}=\sqrt{150\times 90\times 50\times 10}{m}^2$

$=\sqrt{5\times 5\times 30\times 30\times 10\times 10\times 3}{m}^2$
$=\sqrt{5^2}\times \sqrt{{30}^2}\times \sqrt{{10}^2}\times \sqrt{3}{m}^2$
$=5\times 30\times 10\sqrt{3}{m}^2$

$=1500\sqrt{3}{m}^2$

Therefore, area of the triangular plot is $1500\sqrt{3}{m}^2$

Hence, Option A is correct.