Question

Q.3. The lengths of the sides of a
triangle are in the ratio 2:3:4 and its
perimeter is 144 cm. Find the area of
the triangle.
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Answer 1

AnswEr :-

• Area of the triangle is 743.431cm².

Given :-

• The lengths of the sides of a triangle are in the ratio 2:3:4 and it's perimeter is 144cm.

To Find :-

• Area of the triangle.

SoluTion :-

Put x in the ratio.

Then,

• 2x
• 3x
• 4x

Here,

• Perimeter of the triangle is 144cm.

We know that the formula of perimeter of the triangle is :-

Sum of all sides

According to question :-

2x + 3x + 4x = 144

→ 5x + 4x = 144

→ 9x = 144

→ x = 144/9

→ x = 16

Put the value of x in the ratio

Now,

Sides are

• 2x = 2 × 16 = 32
• 3x = 3 × 16 = 48
• 4x = 4 × 16 = 64

Semi Perimeter (S) = 144/2 = 72

• Area of the triangle : √ s (s - a) (s - b) (s - c)

According to question :-

→√ 72 (72 - 32) (72 - 48) (72 - 64)

→ √ 72 × 40 × 24 × 8

→ √ 552690

→ 743.431 cm²

Hence, the area of the triangle is 743.431cm².

_____________________

Answer 2

Solution :

$\bf{\red{\underline{\underline{\bf{Given\::}}}}} </p><p></p><p>$

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

$\bf{\red{\underline{\underline{\bf{To\:find\::}}}}} </p><p></p><p>$

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

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

We know that formula of the perimeter of triangle :

$\begin{gathered}\mapsto\sf{\orange{Perimeter\:of\:triangle=Side+Side+Side}}\\\\\mapsto\sf{144cm=2r+3r+4r}\\\\\mapsto\sf{144cm=9r}\\\\\mapsto\sf{r=\cancel{\dfrac{144}{9}} }cm\\\\\mapsto\sf{\pink{r=16\:cm}}\end{gathered} </p><p>$

$\bf{Using\:Heron's\:Formula\::}</p><p>$

$\begin{gathered}\mapsto\sf{\orange{Sem-perimeter=\dfrac{a+b+c}{2} }}\\\\\\\mapsto\sf{Semi-perimeter=\dfrac{32cm+48cm+64cm}{2} }\\\\\\\mapsto\sf{Semi-perimeter=\cancel{\dfrac{144}{2}} cm}\\\\\\\mapsto\sf{\pink{Semi-perimeter=72cm}}\end{gathered} </p><p></p><p>$

So;

$</p><p>\begin{gathered}\mapsto\sf{Area\:_{triangle}=\sqrt{s(s-a)(s-b)(s-c)} }\\\\\mapsto\sf{Area\:_{triangle}=\sqrt{72(72-32)(72-48)(72-64)}}\\ \\\mapsto\sf{Area\:_{triangle}=\sqrt{72(40)(24)(8)} }\\\\\mapsto\sf{Area\:_{traingle}=\sqrt{552960} cm^{2} }\\\\\mapsto\sf{\pink{Area\:_{triangle}=743.61\:cm^{2} }}\end{gathered} </p><p></p><p>$

Now;

$\begin{gathered}\leadsto\sf{\orange{Area\:of\:triangle=\frac{1}{2} \times base\times height}}\\\\\\\leadsto\sf{743.61cm^{2} =\dfrac{1}{\cancel{2}} \times \cancel{64}cm\times h}\\\\\\\leadsto\sf{743.61\:cm^{2} =32\times h}\\\\\\\leadsto\sf{h=\cancel{\dfrac{743.61cm^{2} }{32cm} }}\\\\\\\leadsto\sf{\pink{h=23.23\:cm}}\end{gathered} </p><p>$