Question

If center of a regular hexagon is at the origin and one of the vertices on the Argand diagram is $1+2i$, then its perimeter is

• A. $2\sqrt{5}$
• B. $6\sqrt{2}$
• C. $4\sqrt{5}$
• D. $6\sqrt{5}$

Answer 1

Answer: (D)

$6\sqrt{5}$

Let the center be denoted by $C(0,0)$.
And let A be one of the vertices.
Hence
$A=1+2i$
Now length of a side of regular hexagon is half of the total length of a diagonal. ...(by property of hexagon).
Length of $CA$
$=\sqrt{1+4}$
$=\sqrt{5}$
Now CA is half the length of the diagonal.
Thus length of a side of this regular polygon
$=CA$
$=\sqrt{5}$
$=a$
Now perimeter is $6a$
$=6\left(\sqrt{5}\right)$