Question

The side of a triangle are in the ratio 13:14:15 and its perimeter is 84cm. Find the area of triangle sides.

Answer 1

$\bf{\huge{\underline{\boxed{\sf{\blue{Answer:}}}}}}$

$\bf{\underline{\underline{\tt{\purple{Given:}}}}}$

• The sides of a triangle are in the ratio 13:14:15
• Perimeter of the triangle is 84 cm

$\bf{\underline{\underline{\tt{\purple{To\:find:}}}}}$

• Area of the triangle

$\bf{\underline{\underline{\tt{\purple{Solution:}}}}}$

Let x be the common multiple for the ratio of the sides of the triangle.

•°• First side = a = 13x

Second side = b = 14x

Third side = c = 15x

Using the formula for perimeter of a triangle, we can further solve the question and thereby derive the value of x.

$\bf{\large{\underline{\boxed{\sf{\green{Perimeter\:of\:a\:triangle\:=\:a\:+\:b\:+\:c\:}}}}}}$

Plug in the values,

=> 84 = 13x + 14x + 15x

=> 84 = 27x + 15x

=> 84 = 42x

=> $\large\frac{84}{42}$ = x

=> x = 2

Substitute x = 2 in the values of the ratio of sides of the triangle.

$\bf{\large{\underline{\boxed{\sf{\blue{First\:side\:=\:a=\:13x\:=\:13\times\:2=\:26\:cm}}}}}}$

$\bf{\large{\underline{\boxed{\sf{\blue{Second\:side\:=\:b=\:14x\:=\:14\times\:2=\:28\:cm}}}}}}$

$\bf{\large{\underline{\boxed{\sf{\blue{Third\:side\:=\:c=\:15x\:=\:1</p><p>5\times\:2=\:30\:cm}}}}}}$

Now, to find the area of the triangle, we will use herons formula. For this we need to find the semiperimeter "s"

=> Semiperimeter,s = $\large\frac{Perimeter}{2}$

=> Semiperimeter,s = $\large\frac{84}{2}$

=> Semiperimeter, s = 42

Let's move to heron's formula,

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

Substitute the appropriate values of s, a, b and c in the formula.

=$\sqrt{42(42-26) (42-28) (42-30)}$

= $\sqrt{42(16) (14) (12)}$

= $\sqrt{42\times\:224\times\:12}$

= $\sqrt{42\times\:2688}$

= $\sqrt{112896}$

=> Area = 336 sq.cm

$\bf{\large{\underline{\boxed{\sf{\orange{Area\:of\:triangle\:=\:336\:cm^2}}}}}}$

Answer 2

ANSWER:-

Given:

The sides of a triangle are in the ratio 13:14:15 & its perimeter is 84cm.

To find:

Find the area of triangle sides

Solution:

Let the number be x.

The sides of a ∆ are;

⚫A=13x

⚫B=14x

⚫C=15x

⚫Perimeter= 84cm

We know that, perimeter of triangle is;

=) side + side + side

=) 13x + 14x + 15x = 84

=) 42x= 84

=) x= 84/42

=) x= 2cm

So,

1st side, 13x = 13×2 = 26cm

2nd sise, 14x= 14×2 = 28cm

3rd side, 15x= 15× 2 = 30cm

Now,

We using Heron's Formula:

$s = \frac{a + b + c}{2} \\ \\ = > \frac{26 + 28 + 30}{2} \\ \\ = > \frac{84}{2} \\ \\ = > 42cm$

Therefore,

Area of triangle:

$A= \sqrt{s(s - a)(s - b)(s - c)} \\ \\ = > \sqrt{42(42 - 26)(42 - 28)(42 - 30)} \\ \\ = > \sqrt{42(16)(14)(12)} \\ \\ = > \sqrt{2 \times 3 \times7 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7 \times 2 \times 2 \times 3 } \\ \\ = > 2 \times 2 \times 2 \times 2 \times 3 \times 7 \\ \\ = > 336 {cm}^{2}$

Hence,

Area of ∆ is 336cm².

Hope it helps ☺️