What is the area of a parallelogram whose lengths of two adjacent sides are 8 cm and12 cm, and the length of one of its diagonals is 16 cm?

Let ABCD be the parallelogram.

It is known that the diagonal of a parallelogram divides it into triangles with equal areas.

∴ Area of parallelogram ABCD = 2 × Area (ΔABC) ... (1)

Let a = 12 cm, b = 8 cm and c = 16 cm

∴ Semi-perimeter, $\mathrm{s}=\frac{12+8+16}{2}=18 \mathrm{cm}$

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

$$\begin{gathered} =\sqrt{18(18-12)(18-8)(18-16)} \\ =12\sqrt{15} {\mathrm{cm}}^2 \end{gathered}$$

From equation (1),

Area of parallelogram ABCD = $2\times 12\sqrt{15}=24\sqrt{15} {\mathrm{cm}}^2$