The area of a trapezium is 384 cm

Given:

Area of the trapezium $=384 \mathrm{~cm}^{2}$

The parallel sides are in the ratio $3: 5$ and the perpendicular height between them is $12 \mathrm{~cm}$.

Suppose that the sides are in $\mathrm{x}$ multiples of each other.

Then, length of the shorter side $=3 \mathrm{x}$

Length of the longer side $=5 \mathrm{x}$

Area of a trapezium $=\frac{1}{2} \times($ Sum of parallel sides $) \times($ Height $)$

$\Rightarrow 384=\frac{1}{2} \times(3 \mathrm{x}+5 \mathrm{x}) \times(12)$

$\Rightarrow 384=\frac{12}{2} \times(8 \mathrm{x})$

$\Rightarrow 384=6 \times(8 \mathrm{x})$

$\Rightarrow 8 x=\frac{384}{6}=64$

$\Rightarrow x=\frac{64}{8}=8 \mathrm{~cm}$

$\therefore$ Length of the shorter side $=3 \times \mathrm{x}=3 \times 8=24 \mathrm{~cm}$

And, length of the longer side $=5 \times \mathrm{x}=5 \times 8=40 \mathrm{~cm}$