Question

Sides of a triangle are in the ratio 12:17:25 and its perimeter is 540 cm. Find its Area .​

Answer 1

Answer:

Given :-

• The sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm.

To Find :-

• What is the area.

Formula Used :-

$\clubsuit$ Area Of Triangle by using Heron's formula :

$\footnotesize \bigstar \: \: \sf\boxed{\bold{\pink{Area_{+Triangle)} =\: \sqrt{s(s - a)(s - b)(s - c)}}}}\: \: \: \bigstar$

where,

• s = Semi-perimeter of a triangle
• a = First Side Of Triangle
• b = Second Side Of Triangle
• c = Third Side Of Triangle

Solution :-

Let,

$\mapsto \bf First\: Side_{(Triangle)} =\: 12x$

$\mapsto \bf Second\: Side_{(Triangle)} =\: 17x$

$\mapsto \bf Third\: Side_{(Triangle)} =\: 25x$

As we know that :

$\footnotesize \bigstar \: \: \sf\boxed{\bold{\pink{Perimeter_{(Triangle)} =\: Sum\: of\: all\: sides}}}\: \: \: \bigstar$

According to the question by using the formula we get,

$\implies \bf Perimeter_{(Triangle)} =\: a + b + c$

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

$\implies \sf 29x + 25x =\: 540$

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

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

$\implies \sf\bold{\blue{x =\: 10\: cm}}$

Hence, the required sides of a triangle are :

☆ First Side Of Triangle :

➳ First Side Of Triangle = 12x

➳ First Side Of Triangle = 12 × 10

➳ First Side Of Triangle = 120 cm

☆ Second Side Of Triangle :

➳ Second Side Of Triangle = 17x

➳ Second Side Of Triangle = 17 × 10

➳ Second Side Of Triangle = 170 cm

☆ Third Side Of Triangle :-

➳ Third Side Of Triangle = 25x

➳ Third Side Of Triangle = 25 × 10

➳ Third Side Of Triangle = 250 cm

Now, we have to find the semi-perimeter of a triangle :

Given :

• First Side (a) = 120 cm
• Second Side (b) = 170 cm
• Third Side (c) = 250 cm

As we know that :

$\footnotesize \bigstar \: \: \sf\boxed{\bold{\pink{Semi-Perimeter_{(Triangle)} =\: \dfrac{Sum\: of\: all\: sides}{2}}}}\: \: \: \bigstar$

According to the question by using the formula we get,

$\implies \bf Semi-Perimeter =\: \dfrac{a + b + c}{2}$

$\implies \sf Semi-Perimeter =\: \dfrac{120 + 170 + 250}{2}$

$\implies \sf Semi-Perimeter =\: \dfrac{\cancel{540}}{\cancel{2}}$

$\implies \sf\bold{\purple{Semi-Perimeter =\: 270\: cm}}$

Now, we have to find the area of a triangle :

Given :

• Semi-Perimeter (s) = 270 cm
• First Side (a) = 120 cm
• Second Side (b) = 170 cm
• Third Side (c) = 250 cm

According to the question by using the Heron's formula we get,

$\implies \bf Area_{(Triangle)} =\: \sqrt{s(s - a)(s - b)(s - c)}$

$\implies \sf Area_{(Triangle)} =\: \sqrt{270(270 - 120)(270 - 170)(270 - 250)}$

$\implies \sf Area_{(Triangle)} =\: \sqrt{270(150)(100)(20)}$

$\implies \sf Area_{(Triangle)} =\: \sqrt{270(300000)}$

$\implies \sf Area_{(Triangle)} =\: \sqrt{270 \times 300000}$

$\implies \sf Area_{(Triangle)} =\: \sqrt{81000000}$

$\implies \sf\bold{\red{Area_{(Triangle)} =\: 9000\: cm^2}}$

$\therefore$ The area of a triangle is 9000 cm² .

Answer 2

☯ Given :-

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

$\:$

☯ To Find :-

Area of the triange

$\:$

☯ Used Formulas :-

$\red⇝ \: \bf Perimeter _{(Triange)} = a +b+c$

$\displaystyle \red⇝ \: \bf Area _{(Triange)} = \sqrt{s(s - a)(s - b)(s - c) \: }$

$\:$

☯ Solution :-

Sides of a Δle are in ratio = 12:17:25

Those perimeter is 540cm

$\displaystyle\bf ⟼ \: Semi-perimeter(s) = \cancel \frac{540}{2} = 270 cm$

$\purple⟶ \: \bf Perimeter _{(Triange)} = a +b+c$

$\purple ⟶ \bf \: 540 = 12x+17x+25x$

$\purple ⟶ \bf \: 540 = 54x$

$\displaystyle\purple ⟶ \bf \: \cancel\frac{540}{54} = x$

$\purple ⟶ \: \red{ \boxed{ \bf x= 10}}$

∴ 12x = 10×12 = 120

∴ 17x = 10×17 = 170

∴ 25x = 10×25 = 250

The sides of the triange are 120, 170 and 250

$\:$

☯ Proof :-

$\purple ⟶ \bf \: 540 = 120+170+250$

$\purple ⟶ \bf \: 540 = 540$

∴ Hence, proved

$\:$

By using Heron’s formula,

$\bf \green⟶ \: Area_{(Triangle)}= \sqrt{s(s - a)(s - b)(s - c) \: }$

$\bf \green⟶ \: Area_{(Triangle)}= \sqrt{270(270 - 120)(270 - 170)(270 - 250)}$

$\bf \green⟶ \: Area_{(Triangle)}= \sqrt{270(270 - 120)(270 - 170)(270 - 250)} \:$

$\bf \green⟶ \: Area_{(Triangle)} = \sqrt{270 × 150 × 100 × 20}$

$\bf \green⟶ \: Area_{(Triangle)} = \sqrt{81000000}$

$\bf \green⟶ \: Area_{(Triangle)} = \red{9000 cm^{2} }$