Question

The sides of the triangle are in the ratio 2:3:4 and its perimeter is 144cm. find its area.
pls give answer fast​

Answer 1

Answer:

Solution :

\bf{\red{\underline{\underline{\bf{Given\::}}}}}Given:

The length of side of a triangle are in the ratio 2:3:4 and it perimeter is 144 cm.

\bf{\red{\underline{\underline{\bf{To\:find\::}}}}}Tofind:

The area of the triangle and the height corresponding to longest side.

\bf{\red{\underline{\underline{\bf{Explanation\::}}}}}Explanation:

Let the ratio be r

The sides of the triangle :

2r

3r

4r

We know that formula of the perimeter of triangle :

\begin{gathered}\mapsto\sf{\orange{Perimeter\:of\:triangle=Side+Side+Side}}\\\\\mapsto\sf{144cm=2r+3r+4r}\\\\\mapsto\sf{144cm=9r}\\\\\mapsto\sf{r=\cancel{\dfrac{144}{9}} }cm\\\\\mapsto\sf{\pink{r=16\:cm}}\end{gathered}↦Perimeteroftriangle=Side+Side+Side↦144cm=2r+3r+4r↦144cm=9r↦r=9144cm↦r=16cm

Now;

1st side = 2(16) = 32 cm

2nd side = 3(16) = 48 cm

3rd side = 4(16) = 64 cm

\dag\bf{\underline{\underline{\bf{Using\:Heron's\:Formula\::}}}}}

\begin{gathered}\mapsto\sf{\orange{Sem-perimeter=\dfrac{a+b+c}{2} }}\\\\\\\mapsto\sf{Semi-perimeter=\dfrac{32cm+48cm+64cm}{2} }\\\\\\\mapsto\sf{Semi-perimeter=\cancel{\dfrac{144}{2}} cm}\\\\\\\mapsto\sf{\pink{Semi-perimeter=72cm}}\end{gathered}↦Sem−perimeter=2a+b+c↦Semi−perimeter=232cm+48cm+64cm↦Semi−perimeter=2144cm↦Semi−perimeter=72cm

So;

\begin{gathered}\mapsto\sf{Area\:_{triangle}=\sqrt{s(s-a)(s-b)(s-c)} }\\\\\mapsto\sf{Area\:_{triangle}=\sqrt{72(72-32)(72-48)(72-64)}}\\ \\\mapsto\sf{Area\:_{triangle}=\sqrt{72(40)(24)(8)} }\\\\\mapsto\sf{Area\:_{traingle}=\sqrt{552960} cm^{2} }\\\\\mapsto\sf{\pink{Area\:_{triangle}=743.61\:cm^{2} }}\end{gathered}↦Areatriangle=s(s−a)(s−b)(s−c)↦Areatriangle=72(72−32)(72−48)(72−64)↦Areatriangle=72(40)(24)(8)↦Areatraingle=552960cm2↦Areatriangle=743.61cm2

Now;

\begin{gathered}\leadsto\sf{\orange{Area\:of\:triangle=\frac{1}{2} \times base\times height}}\\\\\\\leadsto\sf{743.61cm^{2} =\dfrac{1}{\cancel{2}} \times \cancel{64}cm\times h}\\\\\\\leadsto\sf{743.61\:cm^{2} =32\times h}\\\\\\\leadsto\sf{h=\cancel{\dfrac{743.61cm^{2} }{32cm} }}\\\\\\\leadsto\sf{\pink{h=23.23\:cm}}\end{gathered}⇝Areaoftriangle=21×base×height⇝743.61cm2=21×64cm×h⇝743.61cm2=32×h⇝h=32cm743.61cm2⇝h=23.23cm