Question

The coordinates of two consecutive vertices $A$ and $B$ of a regular hexagon $ABCDEF$ are $(1,0)$ and $(2,0),$ respectively. Then the equation of the diagonal $CE$ is

• A. $\sqrt{3}x+y=4$
• B. $x+\sqrt{3}y+4=0$
• C. $x+\sqrt{3}y=4$
• D. $x-\sqrt{3}y=4$

Answer 1

The correct option is C $x+\sqrt{3}y=4$


Sum of interior angles $=(6-2){180}^{\circ }={720}^{\circ }$
Each interior angle of the hexagon is ${120}^{\circ }.$
$\therefore \angle CBX={60}^{\circ }$
$BC$ makes an angle ${60}^{\circ }$ with $x-$axis.
Length of side of hexagon is $1$ unit.
Using parametric form,
$C\equiv (2+1\cos {60}^{\circ },0+1\sin {60}^{\circ })$
$\therefore C\equiv (\frac{5}{2},\frac{\sqrt{3}}{2})$

Coordinates of $E$ are $(1,2\times \sin {60}^{\circ })=(1,\sqrt{3})$
Hence, equation of the diagonal $CE$ is
$y-\sqrt{3}=\frac{\frac{\sqrt{3}}{2}-\sqrt{3}}{\frac{5}{2}-1}(x-1)$
i.e., $x+\sqrt{3}y=4$