Question

The perimeter of a triangle is 50 cm. One side of a triangle is 4 cm longer than the smaller side and the third side is 6 cm less than twice the smaller side. Find the area of the triangle.

Answer 1

Given: The perimeter of a triangle is 50 cm.

Now, semi-perimeter(s) of the triangle is

= `"Perimeter of triangle"/2`

= `50/2`

= 25

Suppose that the smaller side of the triangle be a = x cm.

So, the second side will be b = (x + 4) cm and 3^rd side will be c = (2x – 6) cm.

Now, perimeter of triangle = a + b + c = x + (x + 4) + (2x – 6)

50 cm = (4x – 2) cm

50 = 4x – 2

4x = 50 + 2

4x = 52

x = `52/4`

x = 13

Since, the three side of the triangle are:

a = x = 13,

b = x + 4 = 13 + 4 = 17

c = 2x – 6 = 2 × 13 – 6 = 26 – 6 = 20.

So, area of the triangle = `sqrt(s(s - a)(s - b)(s - c))`

= `sqrt(25 xx (25 - 13) xx (25 - 17) xx (25 - 20))`

= `sqrt(25 xx 12 xx 8 xx 5)`

= `sqrt(5 xx 5 xx 4 xx 3 xx 4 xx 2 xx 5)`

= `5 xx 4 xx 20sqrt(30)  cm^2`

= `20sqrt(30)  cm^2`

Therefore, the area of a triangle is `20sqrt(30)  cm^2`.