Question

In a rectangle, the angle between a diagonal and a side is 30° and the length of this diagonal is 8 cm. the area of the rectangle is

(a) 16 cm²
(b) $\frac{16}{\sqrt{3}}{\mathrm{cm}}^2$
(c) $16\sqrt{3}{\mathrm{cm}}^2$
(d) $8\sqrt{3}{\mathrm{cm}}^2$

Answer 1

(c) $16\sqrt{3}{\mathrm{cm}}^2$
Let $ABCD$ be the rectangle in which ∠$BAC={30}^o$ and $AC=8$ cm.

In ∆$BAC$, we have:

$\frac{AB}{AC}=\cos {30}^o=\frac{\sqrt{3}}{2}$

⇒ $\frac{AB}{8}=\frac{\sqrt{3}}{2}$
⇒ $AB=8\frac{\sqrt{3}}{2}=4\sqrt{3}$ m
Again,

$\frac{BC}{AC}=\sin {30}^o=\frac{1}{2}$

⇒ $\frac{BC}{8}=\frac{1}{2}$
⇒ $BC=\frac{8}{2}=4$ m
∴ Area of the rectangle = $(AB\times BC)=(4\sqrt{3}\times 4)=16\sqrt{3}$ cm²