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soldier1979 [14.2K]
2 years ago
7

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

Mathematics
2 answers:
STALIN [3.7K]2 years ago
8 0

Given :

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

⠀

To Find :

  • Its area.

⠀

Solution :

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

⠀

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

⠀

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

⠀

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

⠀

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}

kifflom [539]2 years ago
5 0

Answer:

2598m²

Step-by-step explanation:

<em>Figure</em><em> </em><em>:</em><em> </em>

\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

<em>According</em><em> </em><em>to</em><em> </em><em>the</em><em> </em><em>question</em><em> </em><em>,</em>

\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 ,

<em>Heron's</em><em> </em><em>Formula</em><em> </em><em>:</em><em>-</em>

  • 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 !

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