Question

The difference between the semiperimeter and the sides of a ∆ABC are 8 cm, 7 cm and 5 cm respectively. Find the area of the triangle.

Answer 1

Let the semi-perimeter of the triangle be s.
Let the sides of the triangle be a, b and c.
Given: s − a = 8, s − b = 7 and s − c = 5 ....(1)

Adding all three equations, we get
3s − (a + b + c) = 8 + 7 + 5
⇒ 3s − (a + b + c) = 20
⇒ 3s − 2s = 20 $\left(\because s=\frac{a+b+c}{2}\right)$
⇒ s = 20 cm ...(2)

∴ By Heron's formula,
$$\begin{gathered} \mathrm{Area}\mathrm{of}∆=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)} \\ =\sqrt{20\left(8\right)\left(7\right)\left(5\right)}\left(\mathrm{from}\left(1\right)\mathrm{and}\left(2\right)\right) \\ =20\sqrt{14}{\mathrm{cm}}^2 \end{gathered}$$

Hence, the area of the triangle is $20\sqrt{14}{\mathrm{cm}}^2$.