Question

The area of a triangle is 150 sq cm and its sides are in the ratio 3 : 4 : 5. What is its perimeter?

Answer 1

Given :

• Area of the triangle = 150 cm².

• Ratio of sides = 3 : 4 : 5

To find :

The perimeter of the triangle.

Solution :

Let the sides of the triangle (in terms of x) be 3x , 4x and 5x.

Since, the sides are in the different ratio , the triangle is Scalene one.

Formula for area of a Scalene triangle :-

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

Where :-

• A = Area of the triangle
• a,b and c = Side of the triangle
• s = Semi-perimeter

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

Now , first let us find the semi-perimeter here .

By using the formula for Semi-perimeter of a triangle and substituting the values in it, we get :

$:\implies \bf{s = \dfrac{3x + 4x + 5x}{2}}$

$:\implies \bf{s = \dfrac{12x}{2}}$

$:\implies \bf{s = 6x}$

$\underline{\therefore \bf{Semi-perimeter\:(s) = 6x}}$

Hence, the semi-perimeter of the triangle is 6x.

Now , using the formula for area of a Scalene triangle and substituting the values in it, we get :

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

$:\implies \bf{150 = \sqrt{6x(6x - 3x)(6x - 4x)(6x - 5x)}}$

$:\implies \bf{150 = \sqrt{6x \times 3x \times 2x \times x}}$

$:\implies \bf{150 = \sqrt{36x^{4}}}$

$:\implies \bf{150 = \sqrt{6x^{2} \times 6x^{2}}}$

$:\implies \bf{150 = 6x^{2}}$

$:\implies \bf{\dfrac{150}{6} = x^{2}}$

$:\implies \bf{25 = x^{2}}$

$:\implies \bf{\sqrt{25} = x}$

$:\implies \bf{5 = x}$

$\underline{\therefore \bf{x = 5\:cm}}$

Hence, the value of x is 5 cm.

Now by substituting the value of x in the three sides of the triangle in terms of x , we get :

• $:\implies \bf{a = 3x}$

$:\implies \bf{a = 3 \times 5}$

$:\implies \bf{a = 15}$

$\underline{\therefore \bf{a = 15\:cm}}$

Hence, the first side is 15 cm.

• $:\implies \bf{b = 4x}$

$:\implies \bf{b = 3 \times 5}$

$:\implies \bf{b = 20}$

$\underline{\therefore \bf{b = 20\:cm}}$

Hence, the second side is 20 cm.

• $:\implies \bf{c = 5x}$

$:\implies \bf{c = 5 \times 5}$

$:\implies \bf{c = 25}$

$\underline{\therefore \bf{x = 25\:cm}}$

Hence, the third side is 25 cm.

To Find the Perimeter of the triangle :-

By using the formula for perimeter of a triangle , we get :

$\boxed{:\implies \bf{P = Sum\:of\:all\:sides}}$

$:\implies \bf{P = a + b + c}$

$:\implies \bf{P = 15 + 20 + 25}$

$:\implies \bf{P = 60}$

$\underline{\therefore \bf{Perimeter\:(P) = 60\:cm}}$

Hence, the Perimeter of the triangle is 60 cm.

Answer 2

QuestioN :-

The area of a triangle is 150cm² and its sides are in the ratio 3 : 4 : 5. What is its perimeter?

ANsWeR :-

Given :-

• Area of Triangle = 150cm²
• Sides in ratio = 3 : 4 : 5

To find :-

• Perimeter of the triangle

AnsweR :-

Area of triangle using heron's formula

=> A = √s (s - a) (s - b) (s - c)

Semi perimeter = a + b + c / 2

Let the ratios be 3x , 4x , 5x .

=> 3x + 4x + 5x / 2

=> 12x / 2

=> 6x

Heron's formula ,

=> 150 = √6x ( 6x - 3x ) ( 6x - 4x ) ( 6x - 5x )

=> 150 = √6x × 3x × 2x × x

=> 150 = √36x⁴

=> 150 = √ 6x² × 6x²

=> 150 = 6x²

=> 150/6 = x²

=> 25 = x²

=> √25 = x

=> 5 = x

Now , we have found the value of x .

Now , we have to find Perimeter

So ,

3x = 3 × 5 = 15 cm

4x = 4 × 5 = 20 cm

5x = 5 × 5 = 25 cm

Perimeter = Adding all the three sides

=> 15 + 20 + 25

=> 60 cm

Hence , Perimeter of Triangle is 60 cm