The lengths of the three sides of a triangle are 30 cm, 24 cm and 18 cm respectively.

Solution:

(a) 24 cm

Let:

$a=30 \mathrm{~cm}, b=24 \mathrm{~cm}$ and $c=18 \mathrm{~cm}$

$s=\frac{a+b+c}{2}=\frac{30+24+18}{2}=36 \mathrm{~cm}$

On applying Heron's formula, we get:

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

$=\sqrt{36(36-30)(36-24)(36-18)}$

$=\sqrt{36 \times 6 \times 12 \times 18}$

$=\sqrt{12 \times 3 \times 12 \times 6 \times 3}$

$=12 \times 3 \times 6$

$=216 \mathrm{~cm}^{2}$

The smallest side is 18 cm.
Hence, the altitude of the triangle corresponding to 18 cm is given by:

Area of triangle $=216 \mathrm{~cm}^{2}$

$\Rightarrow \frac{1}{2} \times$ Base $\times$ Height $=216$

$\Rightarrow$ Height $=\frac{216 \times 2}{18}=24 \mathrm{~cm}$