Question

the sidesvof a triangle are in the ratio 5:4:3 and perimeter of triangle is 36cm find the area of triangle using heron's formula​

Answer 1

Given :

• Ratio of sides of the triangle = 5 : 4 : 3
• Perimeter of thetriangle = 36 cm

To Find :

• The area of the triangle using heron's formula

Solution :

Let the ratio constant be " x " . Then sides of the triangle are 5x , 4x and 3x.

Perimeter of a triangle is given by ,

$\\ \star \: {\boxed{\purple{\sf{Perimeter_{(triangle)} = a + b + c}}}}$

Here ,

• a , b and c are sides of triangle

Substituting the values we have in the formula ,

$\\ : \implies \sf \: 36 = 5x + 4x + 3x$

$\\ : \implies \sf \: 36 = 12x$

$\\ : \implies \sf \: x = \frac{36}{12}$

$\\ : \implies{\underline{\boxed{\blue{\mathfrak{x = 3}}}}}$

Then , Sides of the given triangle are ;

• 5x = 5(3) = 15 cm
• 4x = 4(3) = 12 cm
• 3x = 3(3) = 9 cm

$\qquad$━━━━━━━━━━━━━━━━━

Now , Area of triangle (heron's formula) is given by ;

$\\ \star \: {\boxed{\purple{\sf{Area_{(triangle)} = \sqrt{s(s - a)(s - b)(s - c)} }}}}$

Here ,

• s is semiperimeter
• a , b and c are sides of triangle

Now , Calculating the semiperimeter of given triangle ;

$\\ : \implies \sf \: s = \frac{15+9+12}{2}$

$\\ : \implies \sf \: s = \frac{36}{2}$

$\\ : \implies{\underline{\boxed {\blue{\mathfrak{s = 18 \: cm}}}}}$

Substituting the values we have in the heron's formula ;

$\\ : \implies \sf \: Area_{(triangle)} = \sqrt{18(18- 15)(18 - 9)(18 - 12)}$

$\\ : \implies \sf \: Area_{(triangle)} = \sqrt{18(3)(9)(6)}$

$\\ : \implies \sf \: Area_{(triangle)} = \sqrt{2916}$

$\\ : \implies{\underline{\boxed{\pink{\mathfrak{Area_{(triangle)} = 54 \: {cm}^{2} }}}}} \: \bigstar$

Hence ,

• The Area of the given triangle is 54 cm²

Answer 2

Answer:

Given :-

• Perimeter of triangle = 36 cm
• Ratio of their sides 5:4:3

To Find :-

Area of triangle

Solution :-

As we know that Perimeter is sum of all sides

Let the sides be 5x , 4x and 3x

$\sf \implies \: 36 = 5x + 4x + 3x$

$\sf \implies \: 36 = 12x$

$\sf \implies \: x = \dfrac{36}{12}$

$\sf \implies \: x = 3$

Sides are

$\sf \: 5x = 5(3) = 15 \: cm$

$\sf \: 4x = 4(3) = 12 \: cm$

$\sf \: 3x = 3(3) = 9 \: cm$

Now,

Let's find Semiperimeter by using Herons formula

$\tt \: semiperimeter \: = \dfrac{perimeter}{2}$

$\tt \: semiperimeter = \cancel \dfrac{36}{2} = 18$

Now,

Let's use Herons formula

$\bf \green{Area = \sqrt{s(s - a)(s - b)(s - c)}}$

$\tt \: Area = \sqrt{18(18 - 15)(18 - 12)(18 - 9)}$

$\tt \: Area = \sqrt{18(3)(6)(9)}$

$\tt Area = 54 \: cm²$