Question

The length of sides of a triangle are in the ratio 17:12:25 and its semiperimeter is 270 cm. Find the area of the triangle.

Answer 1

Let the 1st side (a) of the triangle be 17x cm.
Let the 2nd side (b) of the triangle be 12x cm.
Let the 3rd side (c) of the triangle be 25x cm.
$\therefore$ Semiperimeter (s) of the triangle = $\frac{a+b+c}{2}$
$$\begin{gathered} \Rightarrow 270=\frac{17x+12x+25x}{2} \\ \Rightarrow 270=\frac{54x}{2} \\ \Rightarrow 270=27x \\ \Rightarrow x=\frac{270}{27}=10 \end{gathered}$$

Now, we have the following:
First side (a) = 17x cm = 17 × 10 cm = 170 cm
Second side (b) = 12x cm = 12 × 10 cm = 120 cm
Third side (c) = 25x cm = 25 × 10 cm = 250 cm

By Heron’s formula, we have:

$$\begin{gathered} \mathrm{Area}\mathrm{of}\mathrm{triangle}=\sqrt{s(s-a)(s-b)(s-c)} \\ =\sqrt{270\times (270-170)(270-120)(270-250)} \\ =\sqrt{81000000} \\ =9000{\mathrm{cm}}^2 \end{gathered}$$