Question

The sides of one regular hexagon are larger than that of the other regular hexagon by $1$ cm. If the product of their areas is $243$, then find the side of the smaller regular hexagon.

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

Answer: (A)

$2$

Let sides of the hexagon be $a$ and $b$
Such that, $a-b=1$
Area of hexagon $=\frac{3\sqrt{3}}{2}{s}^2$
$\left(\frac{3\sqrt{3}}{2}{a}^2\right)\left(\frac{3\sqrt{3}}{2}{b}^2\right)=243$
$a^2{b}^2=\frac{243\times 4}{27}$
$b^2(b+1{)}^2=36$
$b(b+1)=6$
$b^2+b-6=0$
$(b+3)(b-2)=0$
$b=2,-3$
Hence, $a=b+1=3$ units
Thus, sides of the hexagon are $2$ units and $3$ units.