Question

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

Answer 1

• The area of the triangle = 336 cm².

Given :

• The ratio of the sides of a triangle = 13 : 14 : 15.
• The perimeter of the triangle = 84 cm.

To Find :

• The area of the triangle.

Solution :

We have to find the area of the triangle.

First, we need to find the sides of the triangle.

Let,

The first side of the triangle be 13x.

The second side of the triangle be 14x.

The third side of the triangle be 15x.

Given,

The perimeter of the triangle is 84 cm.

That means,

$: \implies \rm 13x + 14x + 15x = 84 \: cm \\ \\ : \implies \rm 42x = 84 \: cm \\ \\ : \implies \rm x = \dfrac{84}{42} \: cm \\ \\ : \implies \rm x = 2 \: cm \\ \\ \: \: \rm \therefore \: \: x = 2 \: cm$

So, the sides of the triangle are :

★ The first side = 13x = 13 × 2 cm = 26 cm.

★ The second side = 14x = 14 × 2 cm = 28 cm.

★ The third side = 15x = 15 × 2 cm = 30 cm.

Now, we have to find the area of the triangle.

By heron's formula,

$\blue{\boxed{ \red{ \rm{ \sqrt{s(s - a)(s - b)(s - c)}}}}}$

Where,

• s = semi - perimeter.
• a = first side.
• b = second side.
• c = third side.

We have, the value of a, b and c. So, we need to find the value of s.

For finding the value of s.

Method - 1 :

We know that,

Semi - perimeter is half of the perimeter.

$\: \: \bullet \: \: \rm semi - perimeter = \dfrac{perimeter}{2}$

Given,

The perimeter of the triangle is 84 cm.

$: \implies \rm \dfrac{84}{2} \: cm \\ \\ : \implies \rm 42 \: cm$

Hence, the semi - perimeter is 42 cm.

Method - 2 :

We know that,

$\: \: \bullet \: \: \rm semi - perimeter = \dfrac{perimeter}{2}$

We also know that,

Perimeter of the triangle = a + b + c

We have,

• The value of a = 26 cm.
• The value of b = 28 cm.
• The value of c = 30 cm.

$: \implies \rm \dfrac{a + b + c}{2} \: \: cm \\ \\: \implies \rm \dfrac{26 + 28 + 30}{2} \: \: cm \\ \\ : \implies \rm \dfrac{84}{2} \: \: cm \\ \\ : \implies \rm 42 \: \: cm$

Hence, the semi - perimeter is 42 cm.

Now, substitute all the values of a, b, c and s in the heron's formula.

$: \implies \rm \sqrt{42(42 - 26)(42 - 28)(42 - 30) \: {cm}^{4} } \\ \\ : \implies \rm \sqrt{42(16)(14)(12)} \: {cm}^{2} \\ \\ : \implies \rm \sqrt{(2 \times 3 \times 7)(2 \times 2 \times 2 \times 2)(2 \times 7)(2 \times 2 \times 3)} \: {cm}^{2} \\ \\ : \implies \rm \sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7} \: {cm}^{2} \\ \\ : \implies \rm \sqrt{ {(2)}^{2} \times {(2)}^{2} \times {(2)}^{2} \times {(2)}^{2} \times {(3)}^{2} \times {(7)}^{2} } \: {cm}^{2} \\ \\ : \implies \rm 2 \times 2 \times 2 \times 2 \times 3 \times 7 \: {cm}^{2} \\ \\ : \implies \rm 16 \times 21 \: {cm}^{2} \\ \\ : \implies \rm 336 \: {cm}^{2} \\ \\ \rm \therefore \: \: area \: of \: the \: triangle = 336 \: {cm}^{2}$

Hence,

The area of the triangle is 336 cm².

Answer 2

Question= side of a triangle are in the ratio of 13:14:15 and its perimeter is 84 cm. Find its area?⬇️

Solution=⬇️

Let the sides of the triangle be 13a,14a,15a.

Perimeter of the triangle = Sum of all sides =13a+14a+15a=42a

Given, perimeter of the triangle =84cm

∴42a=84cm

a=2cm

So, the sides of the triangle are 13a=26cm,14a=28cm,15a=30cm.

Area of triangle with sides a, b, and c and s=

2

a+b+c

is

s(s−a)(s−b)(s−c)

.

For triangle with sides 26 cm, 28 cm and 30 cm, s=

2

26+28+30

=42cm

Area of the triangle with 26 cm, 28 cm and 30 cm =

42(42−26)(42−28)(42−30)

=

42×16×14×12

=

6×7×4×4×7×2×6×2

=6×7×4×2=336cm

2