Question

The lengths of the three sides of a triangle are 30 cm, 24 cm and 18 cm respectively. The length of the altitude of the triangle corresponding to the smallest side is
(a) 24 cm
(b) 18 cm
(c) 30 cm
(d) 12 cm

Answer 1

(a) 24 cm

$$\begin{gathered} \mathrm{Let}: \\ a=30\mathrm{cm},b=24\mathrm{cm}\mathrm{and}c=18\mathrm{cm} \\ s=\frac{a+b+c}{2}=\frac{30+24+18}{2}=36\mathrm{cm} \\ \mathrm{On}\mathrm{applying}\mathrm{Heron}\text{'}\mathrm{s}\mathrm{formula},\mathrm{we}\mathrm{get}: \\ \mathrm{Area}\mathrm{of}\mathrm{triangle}=\sqrt{s(s-a)(s-b)(s-c)} \\ =\sqrt{36(36-30)(36-24)(36-18)} \\ =\sqrt{36\times 6\times 12\times 18} \\ =\sqrt{12\times 3\times 12\times 6\times 3} \\ =12\times 3\times 6 \\ =216{\mathrm{cm}}^2 \end{gathered}$$

The smallest side is 18 cm.
Hence, the altitude of the triangle corresponding to 18 cm is given by:
$$\begin{gathered} \mathrm{Area}\mathrm{of}\mathrm{triangle}=216{\mathrm{cm}}^2 \\ \Rightarrow \frac{1}{2}\times \mathrm{Base}\times \mathrm{Height}=216 \\ \Rightarrow \mathrm{Height}=\frac{216\times 2}{18}=24\mathrm{cm} \end{gathered}$$