Question

Find the area of the triangle whose sides are 18 cm, 24 cm and 30 cm. Also, find the height corresponding to the smallest side.

Answer 1

Let the sides of triangle be ​a = 18 cm, b = 24 cm and c = 30 cm.
Let s be the semi-perimeter of the triangle.
$$\begin{gathered} s=\frac{1}{2}(a+b+c) \\ s=\frac{1}{2}(18+24+30) \\ s=36\mathrm{cm} \end{gathered}$$


$$\begin{gathered} \mathrm{Area}\mathrm{of}\text{a t}\mathrm{riangle\; =}\sqrt{s(s-a)(s-b)(s-c)} \\ =\sqrt{36(36-18)(36-24)(36-30)} \\ =\sqrt{36\times 18\times 12\times 6} \\ =\sqrt{46656} \\ =216{\mathrm{cm}}^2 \end{gathered}$$
The smallest side is 18 cm long. This is the base.

Now, area of a triangle =$\frac{1}{2}\times b\times h$
$$\begin{gathered} \Rightarrow 216=\frac{1}{2}\times 18\times h \\ \Rightarrow 216=9h \\ \Rightarrow \frac{216}{9}=h \\ \Rightarrow h=24\mathrm{cm} \end{gathered}$$

The height corresponding to the smallest side is 24 cm.