Question

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

Answer 1

Given :

• The sides of a triangular plot are in the ratio 3 : 5 : 7 .
• Its perimeter is 300 m.

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To Find :

• Its area.

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Solution :

• Let us assume the sides in metres be 3x, 5x and 7x .

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Then, We know,

$\qquad \sf \dashrightarrow \: 3x + 5x + 7x = 300 \: \: \: \: \: \: \: Perimeter_{(Triangle)}$

$\qquad \sf \dashrightarrow \: 15x = 300$

$\qquad \sf \dashrightarrow \: x = \dfrac{300}{15}$

$\qquad \bf\dashrightarrow \: x = 20$

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So, The sides of the triangle are :

$\qquad \sf \dashrightarrow \: 3 \times 20 \: m = \bf60 \: m$

$\qquad \sf \dashrightarrow \: 5 \times 20 \: m = \bf 100 \: m$

$\qquad \sf \dashrightarrow \: 7 \times 20 \: m = \bf140 \: m$

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Now, Using Heron's formula :

We have,

$\qquad \sf \dashrightarrow \: s = ( {60 + 100 + 140}) \: m = 300 \: m$

$\qquad \sf \dashrightarrow \: s = \dfrac {60 + 100 + 140}2 \: m = 150 \: m$

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And the Area will be :

$\qquad \sf \dashrightarrow \: \sqrt{150(150 - 60)(150 - 100)(150 - 140)} \: {m}^{2}$

$\qquad \sf \dashrightarrow \: \sqrt{150 \times 90 \times 50 \times 10} \: \: {m}^{2}$

$\qquad \sf \dashrightarrow \: 1500 \sqrt{3} \: {m}^{2}$

Answer 2

Answer:

2598m²

Step-by-step explanation:

Figure :

$\setlength{\unitlength}{1 cm}\begin{picture}(0,0)\thicklines\qbezier(1, 0)(1,0)(3,3)\qbezier(5,0)(5,0)(3,3)\qbezier(5,0)(1,0)(1,0)\put(2.85,3.2){ \bf A }\put(0.5,-0.3){ \bf C }\put(5.2,-0.3){ \bf B }\put(5,1){ \bf 3x }\put(2.5, - .5){ \bf 7x }\put(.5,1){ \bf 5x }\put(4.5,4){ \bf not \: to \: scale \: }\end{picture}$

Here we are given that the ratio of sides of a triangular plot is 3:5:7 and its perimeter is 300m . We are interested in finding the area of the rectangle . Firstly , let us take the given ratio's HCF be x , then we may write the ratio as ,

$\longrightarrow 3x : 5x : 7x$

According to the question ,

$\longrightarrow 3x + 5x + 7x = 300m$

$\longrightarrow 15x = 300m$

$\longrightarrow x =\dfrac{300m}{15}$

$\longrightarrow x = 20m$

Therefore , the sides will be ,

$\longrightarrow 3x = 3(20m) = \red{60m}$

$\longrightarrow 5x =5(20m)= \red{100m}$

$\longrightarrow 7x =7(20m)=\red{140m}$

Now we may use Heron's Formula to find out the area of triangle as ,

Heron's Formula :-

• If three sides of a ∆ is a , b , c then the area is given by $\sqrt{ s(s-a)(s-b)(s-c)}$ , where s is the semi perimeter .

Here ,

$\longrightarrow s =\dfrac{300m}{2}=150m$

Therefore ,

$\longrightarrow Area =\sqrt{ 150(150-60)(150-100)(150-140)}m^2$

$\longrightarrow Area =\sqrt{ 150 \times 90 \times 50 \times 10}m^2$

$\longrightarrow Area = \sqrt{ 50^2 \times 30^2\times 3}m^2$

$\longrightarrow Area = 50\times 30 \times 1.732 m^2$

$\longrightarrow\underline{\underline{ Area = 2598m^2}}$

And we are done !