Respuesta :
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,
[tex] \qquad \sf \dashrightarrow \: 3x + 5x + 7x = 300 \: \: \: \: \: \: \: Perimeter_{(Triangle)}[/tex]
[tex]\qquad \sf \dashrightarrow \: 15x = 300 [/tex]
[tex]\qquad \sf \dashrightarrow \: x = \dfrac{300}{15} [/tex]
[tex]\qquad \bf\dashrightarrow \: x = 20 [/tex]
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So, The sides of the triangle are :
[tex]\qquad \sf \dashrightarrow \: 3 \times 20 \: m = \bf60 \: m [/tex]
[tex]\qquad \sf \dashrightarrow \: 5 \times 20 \: m = \bf 100 \: m [/tex]
[tex]\qquad \sf \dashrightarrow \: 7 \times 20 \: m = \bf140 \: m [/tex]
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Now, Using Heron's formula :
We have,
[tex]\qquad \sf \dashrightarrow \: s = ( {60 + 100 + 140}) \: m = 300 \: m [/tex]
[tex]\qquad \sf \dashrightarrow \: s = \dfrac {60 + 100 + 140}2 \: m = 150 \: m [/tex]
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And the Area will be :
[tex]\qquad \sf \dashrightarrow \: \sqrt{150(150 - 60)(150 - 100)(150 - 140)} \: {m}^{2} [/tex]
[tex]\qquad \sf \dashrightarrow \: \sqrt{150 \times 90 \times 50 \times 10} \: \: {m}^{2} [/tex]
[tex] \qquad \sf \dashrightarrow \: 1500 \sqrt{3} \: {m}^{2} [/tex]
Answer:
2598m²
Step-by-step explanation:
Figure :
[tex]\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} [/tex]
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 ,
[tex]\longrightarrow 3x : 5x : 7x [/tex]
According to the question ,
[tex]\longrightarrow 3x + 5x + 7x = 300m \\ [/tex]
[tex]\longrightarrow 15x = 300m\\ [/tex]
[tex]\longrightarrow x =\dfrac{300m}{15}\\ [/tex]
[tex]\longrightarrow x = 20m [/tex]
Therefore , the sides will be ,
[tex]\longrightarrow 3x = 3(20m) = \red{60m}\\ [/tex]
[tex]\longrightarrow 5x =5(20m)= \red{100m}\\ [/tex]
[tex]\longrightarrow 7x =7(20m)=\red{140m}[/tex]
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 [tex]\sqrt{ s(s-a)(s-b)(s-c)}[/tex] , where s is the semi perimeter .
Here ,
[tex]\longrightarrow s =\dfrac{300m}{2}=150m [/tex]
Therefore ,
[tex]\longrightarrow Area =\sqrt{ 150(150-60)(150-100)(150-140)}m^2\\ [/tex]
[tex]\longrightarrow Area =\sqrt{ 150 \times 90 \times 50 \times 10}m^2\\ [/tex]
[tex]\longrightarrow Area = \sqrt{ 50^2 \times 30^2\times 3}m^2\\[/tex]
[tex]\longrightarrow Area = 50\times 30 \times 1.732 m^2\\[/tex]
[tex]\longrightarrow\underline{\underline{ Area = 2598m^2}} [/tex]
And we are done !
