Question

The base of an isosceles triangle measures 80 cm and its area is 360 sq.m., Find the perimeter of the triangle.

Answer 1

Base of an isosceles triangle = c = 80 cm
Its area = 360 m²

Let the congruent sides of an isosceles triangle be x m.
i.e., a = b = x m

Using Heron’s formula, we have:

$$\begin{gathered} s=\frac{a+b+c}{2} \\ =\frac{x+x+80}{2} \\ =\frac{2x+80}{2} \\ =x+40 \end{gathered}$$

Area of an isosceles triangle = $\sqrt{s(s-a)(s-b)(s-c)}$

$$\begin{gathered} \Rightarrow 360=\sqrt{(x+40)(x+40-x)(x+40-x)(x+40-80)} \\ \Rightarrow 360=\sqrt{(x+40)\left(40\right)\left(40\right)(x-40)} \\ \Rightarrow 360=40\sqrt{(x^2-{40}^2)} \\ \Rightarrow \frac{360}{40}=\sqrt{x^2-1600} \\ \Rightarrow 9^2=x^2-1600 \\ \Rightarrow x^2=1681 \\ \mathrm{or},x\mathit{}=41\mathrm{m} \end{gathered}$$


Length of the congruent sides of an isosceles triangle = 41 m
∴ Perimeter of an isosceles triangle =Sum of all sides
= (41 + 41 + 80) cm
= 162 cm