Question

The perimeter and area of the rectangle whose 12 cm long diagonal makes an angle of ${30}^{\circ }$ with one of its longer sides will be equal to

• A. $18+12\sqrt{3},18\sqrt{3}c{m}^2$
• B. $12+12\sqrt{3},36\sqrt{3}c{m}^2$
• C. $12+18\sqrt{3},24\sqrt{3}c{m}^2$
• D. $9+18\sqrt{3},27\sqrt{3}c{m}^2$

Answer 1

The correct option is B

$12+12\sqrt{3},36\sqrt{3}c{m}^2$

Given ABCD is a rectangle having diagonal AB = 12 cm, $\angle CAB={30}^{\circ }$ and $\angle B={90}^{\circ }$

In $\Delta ABC$,

$\angle C=180-(90+{30}^{\circ })={60}^{\circ }$

Now, $\Delta ABC$, angles are ${30}^{\circ },{60}^{\circ },{90}^{\circ }$

$\Rightarrow \sin \left(30\right):\sin \left(60\right):\sin \left(90\right)$

$\Rightarrow \frac{1}{2}:\frac{\sqrt{3}}{2}:1$

$\Rightarrow 1:\sqrt{3}:2$

so side BC, AB, AC are in the ratio $1:\sqrt{3}:2$

The corresponding sides of the traingle can be calculated as

$\begin{array}{ccccc}{30}^{\circ }& & {60}^{\circ }& & {90}^{\circ }\\ 1& :& \sqrt{3}& :& 2\\ x& :& x\sqrt{3}& :& 2x\\ BC& & AB& & AC\\ \downarrow & & \downarrow & & \downarrow \\ 6& & 6\sqrt{3}& & 12\end{array}$

$(2x=12,\Rightarrow x=6$)

Hence, $BC=6cm$ $AB=6\sqrt{3}cm$

Perimeter $=2(AB+BC)=2(6+6\sqrt{3})=12+12\sqrt{3}cm$

Area $=AB\times BC=6\sqrt{3}\times 6=36\sqrt{3}c{m}^2$