In a rectangle, if the angle between a diagonal and a side is 30° and the length of the diagonal is 6 cm, then the area of the rectangle is :

1. 9 cm²

2. $9\sqrt{3}$ cm²

3. 27 cm²

4. 36 cm²

Let the base of rectangle be AB and breadth be BC,

Diagonal AC = 6 cm

In triangle ABC,

$\Rightarrow \cos 30^{\circ} = \dfrac{\text{base}}{\text{hypotenuse}} = \dfrac{AB}{AC} \\[1em] \Rightarrow \dfrac{\sqrt3}{2} = \dfrac{AB}{6} \\[1em] \Rightarrow \dfrac{\sqrt3}{2} \times 6 = AB \\[1em] \Rightarrow AB = 3\sqrt3 \text{ cm.}$

Also,

$\Rightarrow \sin 30^{\circ} = \dfrac{\text{Perpendicular}}{\text{hypotenuse}} = \dfrac{BC}{AC} \\[1em] \Rightarrow \dfrac{1}{2} = \dfrac{BC}{6} \\[1em] \Rightarrow \dfrac{1}{2} \times 6 = BC \\[1em] \Rightarrow BC = 3 \text{ cm.}$

We know that,

Area of rectangle = length × width

= $3\sqrt3 \times 3$

= $9\sqrt3$ cm².

Hence, option 2 is the correct option.