In a rectangle ABCD, its diagonal AC = 15 cm and ∠ACD = α. If cot α = $\dfrac{3}{2}$, find the perimeter and the area of the rectangle.

Given,

cot α = $\dfrac{3}{2}$

By formula,

⇒ cosec² α = 1 + cot² α

⇒ cosec² α = 1 + $\Big(\dfrac{3}{2}\Big)^2$

⇒ cosec² α = 1 + $\dfrac{9}{4}$

⇒ cosec² α = $\dfrac{4 + 9}{4}$

⇒ cosec² α = $\dfrac{13}{4}$

⇒ cosec α = $\sqrt{\dfrac{13}{4}} = \dfrac{\sqrt{13}}{2}$.

By formula,

⇒ cosec α = $\dfrac{\text{Hypotenuse}}{\text{Perpendicular}}$

⇒ cosec α = $\dfrac{AC}{AD}$

$\Rightarrow \dfrac{\sqrt{13}}{2} = \dfrac{15}{AD}$

$\Rightarrow AD = \dfrac{15 \times 2}{\sqrt{13}} = \dfrac{30}{\sqrt{13}}$

By formula,

⇒ cot α = $\dfrac{\text{Base}}{\text{Perpendicular}}$

⇒ cot α = $\dfrac{CD}{AD}$

$\Rightarrow \dfrac{3}{2} = \dfrac{CD}{\dfrac{30}{\sqrt{13}}}$

$\Rightarrow CD = \dfrac{3}{2} \times \dfrac{30}{\sqrt{13}} = \dfrac{45}{\sqrt{13}}$

Perimeter = 2(length + breadth) = 2(CD + AD)

= $2\Big(\dfrac{45}{\sqrt{13}} + \dfrac{30}{\sqrt{13}}\Big)$

= $2\Big(\dfrac{75}{\sqrt{13}}\Big)$

= $\Big(\dfrac{150}{\sqrt{13}}\Big)$

= $\Big(\dfrac{150}{\sqrt{13}} \times \dfrac{\sqrt{13}}{\sqrt{13}}\Big)$      [Rationalising]

= $\dfrac{150\sqrt{13}}{13}$ cm.

Area = length × breadth

= CD × AD

= $\dfrac{45}{\sqrt{13}} \times \dfrac{30}{\sqrt{13}} = \dfrac{1350}{13} = 103\dfrac{11}{13}$ cm².

Hence, perimeter = $\dfrac{150\sqrt{13}}{13}$ cm and area = $103\dfrac{11}{13}$ cm².