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

= ${\displaystyle \frac{\text{Perimeter of triangle}}{2}}$

= ${\displaystyle \frac{50}{2}}$

= 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 = ${\displaystyle \frac{52}{4}}$

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 = ${\displaystyle \sqrt{s(s-a)(s-b)(s-c)}}$

= ${\displaystyle \sqrt{25\times (25-13)\times (25-17)\times (25-20)}}$

= ${\displaystyle \sqrt{25\times 12\times 8\times 5}}$

= ${\displaystyle \sqrt{5\times 5\times 4\times 3\times 4\times 2\times 5}}$

= ${\displaystyle 5\times 4\times 20\sqrt{30} c{m}^2}$

= ${\displaystyle 20\sqrt{30} c{m}^2}$

Therefore, the area of a triangle is ${\displaystyle 20\sqrt{30} c{m}^2}$.