Question

If $P$, $Q$, $R$ and $S$ are represented by the complex numbers $4+i$, $1+6i$, $-4+3i$ and $-1-2i$ respectively, then $PQRS$ is a

• A. Rectangle
• B. Square
• C. Rhombus
• D. Parallelogram

Answer 1

Answer: (B)

Square

Given complex numbers are $P(4+i)$, $Q(1+6i)$, $R(-4+3i)$ and $S(-1-2i)$
Now,

$\left|PQ\right|=\sqrt{{(4-1)}^2+{(1-6)}^2}$
$=\sqrt{9+25}=\sqrt{34}$

$\left|QR\right|=\sqrt{{(1+4)}^2+{(6-3)}^2}$
$=\sqrt{25+9}=\sqrt{34}$

$\left|RS\right|=\sqrt{{(-4+1)}^2+{(3+2)}^2}$
$=\sqrt{9+25}=\sqrt{34}$

and $\left|SP\right|=\sqrt{{(-1-4)}^2+{(-2-1)}^2}$
$=\sqrt{25+9}=\sqrt{34}$

$\therefore \left|PQ\right|=\left|QR\right|=\left|RS\right|=\left|SP\right|$
Now,

$\left|PR\right|=\sqrt{{(4+4)}^2+{(1-3)}^2}$
$=\sqrt{64+4}=\sqrt{68}$
and $\left|QS\right|=\sqrt{{(1+1)}^2+{(6+2)}^2}$
$=\sqrt{4+64}=\sqrt{68}$
$\therefore \left|PR\right|=\left|QS\right|$
Hence, it is a square.