Question

Two adjacent sides of a parallelogram are 30 m and 14 m and the diagonal joining the end points of these sides is 40 m. The area of the parallelogram is
(a) 168 m²
(b) 336 m²
(c) 372 m²
(d) 480 m²

Answer 1

(b) 336 m²

Parallelogram ABCD is shown in the figure. Diagonal AC divides the parallelogram into two congruent triangles ABC and ADC.

Now,
$\mathrm{Area}\mathrm{of}\mathrm{parallelogram}ABCD=\mathrm{Ar}\left(∆ABC\right)+\mathrm{Ar}\left(∆ADC\right)$

Because $∆$ABC and $∆$ADC are congruent, $\mathrm{Area}\mathrm{of}\mathrm{parallelogram}ABCD=2\times \mathrm{Ar}\left(∆ABC\right)$.

Using Hero's formula for the area of triangle ABC, we get:

Semiperimeter, s$=\frac{1}{2}\left(a+b+c\right)=\frac{1}{2}\left(30+14+40\right)=42\mathrm{cm}$

Area of triangle ABC of the parallelogram:

$$\begin{gathered} =\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)} \\ =\sqrt{42\left(42-40\right)\left(42-30\right)\left(42-14\right)} \\ =\sqrt{42\times 2\times 12\times 28} \\ =168{\mathrm{m}}^2 \end{gathered}$$

∴ Area of parallelogram ABCD$=2\times 168=336{\mathrm{m}}^2$