Question

The lengths of the sides of a triangle are in the ratio 4 : 5 : 3 and its perimeter is 192 cm. Find its area.

Answer 1

Step-by-step explanation:

perimeter of triangle =side +side+side

192=4x+5x+3x

192=12x

$x = \frac{192}{12}$

x=16

Answer 2

Given :

• Ratio of the lengths of the sides of a triangle = 4 : 5 : 3
• Perimeter of the triangle = 192 cm

To find :

• Area of the triangle

Knowledge required ::

• Formula of perimeter of triangle :-

⠀⠀⠀Perimeter = sum of all sides

• Formula of area of triangle :- Heron's formula = √s(s - a)(s - b)(s - c)

• Formula to calculate semi - perimeter :-
• Semi - perimeter (s) = (a + b + c)/2

[where : a, b and c are the three sides of the triangle.]

or

• Semi - perimeter = Perimeter/2

Solution :

Let the three sides of the triangle be 4x, 5x and 3x.

• First side = 4x
• Second side = 5x
• Third side = 3x

⠀⇒ Perimeter = Sum of all sides

⠀⠀⠀⠀⠀⠀⇒ 192 = 4x + 5x + 3x

⠀⠀⠀⠀⠀⠀⇒ 192 = 12x

⠀⠀⠀⠀⠀⠀⇒ 192/12 = x

⠀⠀⠀⠀⠀⠀⇒ 96/6 = x

⠀⠀⠀⠀⠀⠀⇒ 48/3 = x

⠀⠀⠀⠀⠀⠀⇒ 16 = x

The value of x = 16

• Substitute the value of x in the sides which we have let.

⠀⠀⠀⠀⠀⠀⇒ 4x = 4 × 16 = 64

⠀⠀⠀⠀⠀⠀⇒ 5x = 5 × 16 = 80

⠀⠀⠀⠀⠀⠀⇒ 3x = 3 × 16 = 48

• First side = 64 cm
• Second side = 80 cm
• Third side = 48 cm

⠀⠀⠀⠀⠀⠀⇒ s = 192/2

⠀⠀⠀⠀⠀⠀⇒ s = 96

Semi - perimeter of the triangle = 96 cm

Take,

• a = 64 cm
• b = 80 cm
• c = 48 cm

⇒ Area = √s(s - a)(s - b)(s - c)

⇒ Area = √96(96 - 64)(96 - 80)(96 - 48)

⇒ Area = √96(32)(16)(48)

⇒ Area = 1536

★ Area of the triangle = 1536 cm²