Question

The perimeter of a triangular garden is 900 cm and its sides are in the ratio 3 : 5 :4 . Using heron's formula , the area of the triangular garden is​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

Sides of a triangular garden are in the ratio 3 : 5 : 4.

Let assume that the sides of triangular garden be taken as a, b and c.

So, a : b : c = 3 : 5 : 4

Let assume that

$\begin{gathered}\begin{gathered}\rm :\longmapsto\:\bf\: Sides \: of \: garden \: be \: -\begin{cases} &\sf{a = 3x} \\ &\sf{b = 5x}\\ &\sf{c = 4x} \end{cases}\end{gathered}\end{gathered}$

Now, it is given that

Perimeter of triangular garden = 900 cm

So,

$\rm :\longmapsto\:3x + 5x + 4x = 900$

$\rm :\longmapsto\:12x = 900$

$\rm :\longmapsto\:x = \dfrac{900}{12} = 75$

Hence,

$\begin{gathered}\begin{gathered}\rm :\longmapsto\:\bf\: Sides \: of \: garden \: are \: -\begin{cases} &\sf{a = 3x = 225 \: cm} \\ &\sf{b = 5x = 375 \: cm}\\ &\sf{c = 4x = 300 \: cm} \end{cases}\end{gathered}\end{gathered}$

Now,

We know,

$\rm :\longmapsto\:\underline{\boxed{\sf Semi \ perimeter \: (s) \: =\dfrac{Perimeter}{2}}}$

$\rm :\implies\:s = \dfrac{900}{2} = 450 \: cm$

Now, We know, Area of triangle using Heron's Formula is given by

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

So, on substituting the values of s, a, b and c, we get

$\rm :\longmapsto\:Area = \sqrt{450(450 - 225)(450 - 375)(450 - 300)}$

$\rm \:  =  \:  \: \sqrt{450 \times 225 \times 75 \times 150}$

$\rm \:  =  \:  \: \sqrt{75 \times 3 \times 2 \times 75 \times 3 \times 75 \times 75 \times 2}$

$\rm \:  =  \:  \: 75 \times 75 \times 3 \times 2$

$\rm \:  =  \:  \: 150 \times 225$

$\rm \:  =  \:  \: 33750 \: {cm}^{2}$

Hence,

$\: \: \: \: \underbrace{ \boxed{ \bf{ \: Area_{(triangular \: garden)} \: = \: 33750 \: {cm}^{2}}}}$