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If the area of a regular hexagon is equal to the area of an equilateral triangle of side $12 \; \text{cm},$ then the length, in cm, of each side of the hexagon is

1. $6 \sqrt{6}$
2. $2 \sqrt{6}$
3. $4 \sqrt{6}$
4. $\sqrt{6}$

Let the side of hexagon be $x \; \text{cm}.$

The area of regular hexagon $= 6 \times \frac{\sqrt{3}}{4} x^{2}$

Now, $6 \times \frac{\sqrt{3}}{4} x^{2} = \frac{\sqrt{3}}{4} (12)^{2}$

$\Rightarrow 6x^{2} = 12 \times 12$

$\Rightarrow x^{2} = 24$

$\Rightarrow x = \sqrt{24}$

$\Rightarrow \boxed{ x = 2 \sqrt{6} \; \text{cm}}$

Correct Answer $: \text{B}$

$\textbf{PS:}$ The regular hexagon

The $\triangle \text{ABC}$ are equilateral triangle.

• The area of an equilateral triangle $= \frac{\sqrt{3}}{4} \times \text{(Side)}^{2}$
• The area of regular hexagon $= 6 \times \frac{\sqrt{3}}{4} \times \text{(Side)}^{2}$
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