Question

sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm . find its area​

Answer 1

$\huge\underbrace{\orange{\tt{\diamond\; Solution:}}}$

Given,

Ratio of sides of ∆le = 12:17:25

Let's

First Side ☞ a = 12x

Second Side ☞ b = 17x

Third Side ☞ c = 25x

Perimeter of the Triangle = 540cm

$\boxed{\sf{Perimeter = Sum\;of\;All\;Sides}}$

$→\;{\sf{a+b+c = 540}}$

$→\;{\sf{12x+17x+25x = 540}}$

$→\;{\sf{54x = 540}}$

$→\;{\sf{x = \cancel{\frac{540}{54}}}}$

$→\;{\bf{x = 10cm}}$

◕➜ Length of First Side(a) = $\fbox\red{120cm}$

◕➜ Length of Second(b) = $\fbox\green{170cm}$

◕➜ Length of Thrid Side(c) = $\fbox\pink{250cm}$

___________

Now, Area of ∆le

$\boxed{\sf{Area\;of\;∆le = \sqrt{s(s-a)(s-b)(s-c)}}}$

Where 's' is

$\boxed{\sf{s = \frac{(a+b+c)}{2}}}$

➝ s = (120+170+250)÷2

➝ s = 540÷2

➝ s = 270cm

${→\;\sf{\sqrt{270(270-120)(270-170)(270-250}}}$

$→\;{\sf{\sqrt{270×150×100×20}}}$

$→\;{\sf{\sqrt{81000000}}}$

$→\;{\sf{\sqrt{9000²}}}$

$→\;{\sf{9000cm²}}$

.:. Area of the ∆le ◕➜ $\fbox\purple{9000cm²}$

Hope It Helps You ✌️

Answer 2

Question :

Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. find its area.

Given :

• Ratio of sides of triangle = 12 : 17 : 25
• Perimeter of triangle = 540 cm

To Find :

• Area of triangle = ?

Solution :

As, sides are in ratio of 12 : 17 : 25

So,

• Let first side, a = 12x
• Second side, b = 17x
• Third side, c = 25x

Now, we are given perimeter of triangle = 540 cm.

We know that perimeter of triangle is the sum of all sides of a triangle.

⟹ Perimeter = a + b + c

By filling values :

⟹ 540 cm = 12x + 17x + 25x

⟹ 540 cm = 54x

⟹ x = $\sf \dfrac{540}{54}$ cm

⟹ x = 10 cm

Therefore,

• First side, a = 12x = 12 × 10 cm = 120 cm
• Second side, b = 17x = 17 × 10 cm = 170 cm
• Third side, c = 25x = 25 × 10 cm = 250 cm

Now, we have to find area of triangle.

To find it, let's use Heron's formula :

According to Heron's formula :

$\large \underline{\boxed{ \bf Area \: of \: triangle = \sqrt{s(s-a)(s-b)(s-c)}}}$

Here,

• $\sf s \: is \: Semi-perimeter = \dfrac{Perimeter}{2} = \dfrac{540 \: cm}{2} = 270 \: cm$
• a is first side = 120 cm
• b is second side = 170 cm
• c is third side = 250 cm

So, by filling values, we have :

$\sf : \implies Area \: of \: triangle = \sqrt{270 \: cm (270 \: cm - 120 \: cm)(270 \: cm - 170 \: cm)(270 \: cm - 250 \: cm)}$

$\sf : \implies Area \: of \: triangle = \sqrt{270 \: cm (150 \: cm)(100 \: cm)(20 \: cm)}$

$\sf : \implies Area \: of \: triangle = \sqrt{270 \: cm \times 150 \: cm \times 100 \: cm \times 20 \: cm}$

$\sf : \implies Area \: of \: triangle = \sqrt{(270 \times 150 \times 100 \times 20) cm^{2} \times cm^{2}}$

$\sf : \implies Area \: of \: triangle = \sqrt{(3 \times 3 \times 3 \times 10 \times 3 \times 5 \times 10 \times 10 \times 10 \times 2 \times 2 \times 5) cm^{2} \times cm^{2}}$

$\sf : \implies Area \: of \: triangle = \sqrt{(\underbrace{3 \times 3} \times \underbrace{3 \times 3} \times \underbrace{5 \times 5} \times \underbrace{10 \times 10} \times \underbrace{10 \times 10} \times \underbrace{2 \times 2} ) \underbrace{cm^{2} \times cm^{2}}}$

$\sf : \implies Area \: of \: triangle = (3 \times 3 \times 5 \times 10 \times 10 \times 2) cm^{2}$

$\sf : \implies Area \: of \: triangle = 9000 cm^{2}$

Hence, Area of triangle is 9000 cm².