Question

In the figure, if LK=$\sqrt{3}$, find MK, ML, KN, MN and perimeter of $\square MNKL$.

• A. $6(2\sqrt{2}+3+\sqrt{3})$ cm
• B. $6(3+\sqrt{3})$ cm
• C. $6(2\sqrt{2}+3)$ cm
• D. $(2\sqrt{2}+3+\sqrt{3})$ cm

Answer 1

Answer: (D)

$(2\sqrt{2}+3+\sqrt{3})$ cm

In $△LKM$

$\tan {30}^{\circ }=\frac{ML}{LK}\Rightarrow ML=\sqrt{3}\times \frac{1}{\sqrt{3}}=1$

$\cos {30}^{\circ }=\frac{LK}{MK}\Rightarrow MK=\frac{LK}{\cos {30}^{\circ }}=\frac{\sqrt{3}}{\frac{\sqrt{3}}{2}}=2$

$NK=MK$ (Side opposite to equal angles )

$\therefore NK=2$

In$△MKN$

$\sin {45}^{\circ }=\frac{NK}{MN}$

$NK=2\sqrt{2}$

Perimeter of $\square MNKL$ $=MN+NK+NL+ML$

$\Rightarrow$Perimeter $=2\sqrt{2}+2+\sqrt{3}+1$

$=(2\sqrt{2}+3+\sqrt{3})cm$

So option $D$ is correct