CAT 2017 Set-2 | Question: 81

Question

Let $\text{ABCDEF}$ be a regular hexagon with each side of length $1$ cm. The area (in sq cm) of a square with $\text{AC}$ as one side is 

• A. $3\sqrt{2}$
• B. $3$
• C. $4$
• D. $\sqrt{3}$

Answer 1

Let's first draw the diagram.

$\text{ABCDEF}$ is a regular hexagon.

So, each angle in the hexagon $= (\frac{n-2}{n})\times180^\circ$ ,  where $n=$number of sides

$\qquad  \qquad = \left(\frac{6-2}{6}\right)\times180^\circ = \left(\frac{4}{6}\right)\times180^\circ = 120^\circ$

Now,  $\frac{AP}{AB}= \cos 30^\circ$

$\Rightarrow \dfrac{\frac{AC}{2}}{1}=\frac{\sqrt{3}}{2}$

$\Rightarrow \frac{AC}{2}=\frac{\sqrt{3}}{2}$

$\Rightarrow \boxed{AC=\sqrt{3}\;\text{cm}}$

$\therefore$  The area of a square with $AC$ as one side$= (AC)^{2}= (\sqrt{3})^{2}=3\;\text{cm}^{2}.$

Correct Answer $:\text{B}$