Question

The sides of a triangle are in the ratio of12:17:25 and its perimeter is 540 cm. find its area:
a)6000
b)9000
c)12000
d)None of these

Answer 1

⇝ Given :-

For A Triangle ;

• Sides are in Ratio 12 : 17 : 25

• Perimeter = 540 cm

⇝ To Find :-

• Area of the Triangle.

⇝ Solution :-

As Sides are in Ratio 12 : 17 : 25 ;

Let,

• First Side = a = 12x

• Second Side = b = 17x

• Third Side = c = 25x

❒ Finding All Sides :-

We Know,

$\rm Perimeter = Sum \: of \: All \: Sides$

$\rm:\longmapsto540 = 12x + 17x + 25x$

$\rm:\longmapsto 54x = 540$

$\purple{ \large :\longmapsto \underline {\boxed{{\bf x = 10} }}}$

Therefore,

• First Side = a = 120 cm

• Second Side = b = 170 cm

• Third Side = c = 250 cm

Also

• Semi Perimeter = s = 270 cm

❒ Finding Area :-

To Calculate Area when three sides are given we will use Heron's Formula which is :

$\bf \red\bigstar \: \: \orange{ \underbrace{ \underline{Area = \sqrt{s(s - a)(s - b)(s - c)} }}}$

where

• a , b and c are sides of Triangle

• s = Semi - Perimeter

⏩ Putting Values In Formula ;

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

$= \sqrt{270 \times 150 \times 100 \times 20 \: }$

$= \sqrt{81000000 \: }$

$\purple{ \large :\longmapsto \underline {\boxed{{\bf Area=9000 \: cm^2} }}}$

Hence,

$\large\underline{\pink{\underline{\frak{\pmb{\text Area\:of\:\text Triangle = 9000 \: {cm}^{2} }}}}}$

Therefore ,

$\Large\underline{\green{\underline{\frak{\pmb{Option \: B \: is \: Correct}}}}}$

Answer 2

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 of triangle.

Solution :-

First, we have to find the sides of all triangle :

Let,

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

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

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

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,

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

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

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

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

$\implies \sf a =\: \dfrac{10}{1}$

$\implies \sf\bold{\purple{a =\: 10}}$

Hence, the required sides of a triangle are :

❒ First Side Of Triangle :

$\implies \sf First\: Side_{(Triangle)} =\: 12a$

$\implies \sf First\: Side_{(Triangle)} =\: (12 \times 10)\: cm$

$\implies \sf\bold{\green{First\: Side_{(Triangle)} =\: 120\: cm}}$

❒ Second Side Of Triangle :

$\implies \sf Second\: Side_{(Triangle)} =\: 17a$

$\implies \sf Second\: Side_{(Triangle)} =\: (17 \times 10)\: cm$

$\implies \sf\bold{\green{Second\: Side_{(Triangle)} =\: 170\: cm}}$

❒ Third Side Of Triangle :

$\implies \sf Third\: Side_{(Triangle)} =\: 25a$

$\implies \sf Third\: Side_{(Triangle)} =\: (25 \times 10)\: cm$

$\implies \sf\bold{\green{Third\: Side_{(Triangle)} =\: 250\: cm}}$

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

As we know that :

$\footnotesize\bigstar\: \: \sf\boxed{\bold{\pink{Semi-Perimeter_{(Triangle)} =\: \dfrac{Sum\: Of\: All\: Sides}{2}}}}\: \: \bigstar$

Given :

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

According to the question by using the formula we get,

$\small\leadsto \bf Semi-Perimeter_{(Triangle)} =\: \dfrac{a + b + c}{2}$

$\small\leadsto \sf Semi-Perimeter_{(Triangle)} =\: \dfrac{120 + 170 + 250}{2}$

$\small\leadsto \sf Semi-Perimeter_{(Triangle)} =\: \dfrac{\cancel{540}}{\cancel{2}}$

$\small\leadsto \sf Semi-Perimeter_{(Triangle)} =\: \dfrac{270}{1}$

$\small\leadsto \sf\bold{\blue{Semi-Perimeter_{(Triangle)} =\: 270\: cm}}$

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

As we know that :

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

Given :

• Semi-Perimeter = 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,

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

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

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

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

$\small\longrightarrow \sf Area_{(Triangle)} =\: \sqrt{\underline{9000 \times 9000}}$

$\small\longrightarrow \sf\bold{\red{Area_{(Triangle)} =\: 9000\: cm}}$

${\small{\bold{\underline{\therefore\: The\: area\: of\: the\: triangle\: is\: 9000\: cm\: .}}}}$

Hence, the correct options is option no (b) 9000 cm .