Question

find the area of a triangle whose sides are in ratio 5 : 12 : 13 and its perimeter is 60 CM in process tell me fast........

Answer 1

$\underline{\huge{\textbf{Solution}}}$

$\underline{\large{\textbf{Given that}}}$

Sides of a triangle are in ratio of 5:12:13 and it's perimeter is 60cm and now we need to find its area ...

So to Find the area follow the simple steps

$\underline{\textbf{Step-1 ) Find The Measure Of Sides..}}$

Now in Question it's said That the sides are in ratio of 5 : 12 : 13  and it's perimeter = 60cm

So we know that

$\textbf{Perimeter ( Of Triangle ) = Sum of all three sides}$

So here Let

$\textbf{X be the constant of ratio}$

Therefore

$\textbf{Measure Of first side ( a )= 5 \times X \implies 5x }$

$\textbf{Measure Of second side ( b ) = 12 \times X \implies 12x }$

$\textbf{Measure Of third side ( c ) = 13 \times X \implies 13x }$

Now according to question ..

$\mathbf{ 5x\: +\: 12x\: + \: 13x= 60cm}$

That is

$\mathbf{ \implies\: 30x\:=\:60cm}$

$\mathbf{ \implies\: x \: = \:\dfrac{60cm}{30}}$

$\mathbf{ \implies\: x \: = \:\dfrac{60cm}{30}}$

$\mathbf{\implies \: X \:=\: 2cm}$

$\textbf{Now Putting Value of X = 2 in the assumed measures of side we have }$

$\textbf{Measure Of first side ( a ) = 5x \implies 5 \times 2 = 10cm }$

$\textbf{Measure Of second side ( b ) = 12x \implies 12 \times 2 = 24cm }$

$\textbf{Measure Of third side ( c ) = 13x \implies 13\times2 = 26cm }$

$\underline{\textbf{Step-2 ) Find the required area }}$

According to $\underline{\textbf{Heron's Formula}}$ We know that area of Triangle =

$\large \boxed{\mathbf{ \sqrt{s(s - a)(s - b)(s - c)}}}$

Now Putting Values Of a , b and c we have

$\mathbf{ \large{ Area \: of \: triangle\:=\:\sqrt{s(s - 10)(s - 24)(s - 26)}}}$

Here ,

$\mathsf{S\:= \:Semi-Perimeter \: = \dfrac{a\:+\:b\:+\:c}{2}}$

$\implies\: \mathsf{S\: = \dfrac{24+26+10}{2}}$

$\implies \mathsf{S \: =\: \dfrac{60}{2} \:= \:30cm }$

Now Putting the Value Of S = 30cm In The Obtained formula we have

$\mathbf{ \large{ Area \: of \: triangle\:=\:\sqrt{30(30 - 10)(30- 24)(30- 26)}}}$

That is

$\mathbf{\large{\implies \:\sqrt{30(30 - 10)(30- 24)(30- 26)}}}$

$\mathbf{\large{\implies \:\sqrt{30(20)(6)(4)}}}$

$\mathbf{\large{\implies \:\sqrt{600 \times 6 \times 4}}}$

$\mathbf{\large{\implies \:\sqrt{14,400cm^4}}}$

Therefore --

$\textbf{Area of Triangle = }$

$\mathbf{\large{\sqrt{14,400cm^4}} \:= \: 120 \: cm^2}$

$\underline{\textbf{Hence the required answer is --}}$

$\boxed{\boxed{\mathbf{\bigstar \: \: Answer\: =\:120\: cm^2}}}$

Answer 2

ANSWER:-

Given:

A triangle whose sides are in ratio 5:12:13 and its perimeter is 60cm.

To find:

Find the area of triangle.

Solution:

Let the ratio of side be x.

So,

5x :12x:13x

We know that, perimeter of triangle;

⏺️Perimeter= 60cm

=) Side + Side + Side

=) 5x + 12x + 13x = 60

=) 30x = 60

=) x= 60/30

=) x= 2cm

Therefore,

⏺️1st side, 5x = (5×2)cm= 10cm

⏺️2nd side,12x=(12×2)cm= 24cm

⏺️3rd side, 13x=(13×2)cm= 26cm

Now,

Using the Heron's Formula, we get;

Here,

⏺️a= 10cm

⏺️b= 24cm

⏺️c= 26cm

$s = \frac{a + b + c}{2} \\ \\ = > \frac{10 + 24 + 26}{2} \\ \\ = > \frac{60}{2} \\ \\ = > 30$

So,

Area of Triangle:

$A= \sqrt{s(s - a)(s - b)(s - c)} \\ \\ = > \sqrt{30(30 - 10)(30 - 24)(30 - 26)} \\ \\ = > \sqrt{30(20)(6)(4)} \\ \\ = > \sqrt{ 2 \times 3 \times 5 \times 2 \times 2 \times 5 \times 2 \times 3 \times 2 \times 2} \\ \\ = > 2 \times 2 \times 2 \times 3 \times 5 \\ \\ = > 120 {cm}^{2}$

Hence,

The area of triangle is 120cm².

Hope it helps ☺️