Question

＠QualityQuestion

Sides of a triangle are in the ratio of $\rm 12:17:25$, and, its perimeter is 540 cm. Find its $\rm area$.

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

• Sides of a triangle are in the ratio of 12 : 17 : 25

Let assume that

Sides of triangle is represented as a, b, c respectively.

So,

• a = 12x

• b = 17x

• c = 25x

Further, given that

Perimeter of triangle = 540 cm

We know, Perimeter of a triangle is defined as sum of a three sides.

So,

$\rm :\longmapsto\:12x + 17x + 25x = 540$

$\rm :\longmapsto\:54x = 540$

$\bf\implies \:x = 10$

Thus,

• a = 120 cm

• b = 170 cm

• c = 250 cm

Now,

$\underline{\boxed{\sf Semi \: Perimeter \ of \ a \ triangle,s= \dfrac{1}{2} (a+b+c)}}$

$\rm \implies\:s = \dfrac{540}{2} = 270 \: cm$

Now,

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

So, on substituting the values, we get

$\red{\rm :\longmapsto\:Area \: of \: triangle}$

$\rm \:  =  \: \sqrt{270(270 - 120)(270 - 170)(270 - 250)}$

$\rm \:  =  \: \sqrt{270(150)(100)(20)}$

$\rm \:  =  \: 100 \sqrt{27(15)(10)(2)}$

$\rm \:  =  \: 100 \sqrt{(3 \times 3 \times 3)(5 \times 3)(5 \times 2)(2)}$

$\rm \:  =  \: 100 \times 2 \times 3 \times 3 \times 5$

$\rm \:  =  \: 9000 \: {cm}^{2}$

Answer 2

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.

$: \implies \bf \: Perimeter = a + b + c$

By filling values :

$: \implies \sf \: 540 cm = 12x + 17x + 25x$

$: \implies \sf\: 540 \: cm = 54 \: x$

$: \implies \sf \: x = \sf \dfrac{540}{54}$

$: \implies \sf \: x = 10 cm$

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 :

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

Here,

$\sf \: s \: \: is \: \: Semi-perimeter = \dfrac{Perimeter}{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \: = \dfrac{540 \: cm}{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \: = 270 \: cm$

• a is first side = 120 cm

• b is second side = 170 cm

• c is third side = 250 cm

So, by filling values, we have :

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

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

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

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

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

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

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

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

Hence, Area of triangle is 9000 cm².