Q. If $\mathcal{z}_{1},\ \mathcal{z}_{2},\ \mathcal{z}_{3},\ \mathcal{z}_{4}$ are four complex numbers represented by the vertices of a quadrilateral taken in order such that $\mathcal{z}_{1} - \mathcal{z}_{4} = \mathcal{z}_{2} - \mathcal{z}_{3}\ \ and\ \arg{\left( \frac{\mathcal{z}_{4} - \mathcal{z}_{1}}{\mathcal{z}_{2} - \mathcal{z}_{1}} \right) = \frac{\pi}{2},}\$then the quadrilateral is
A) A square
B) A rectangle
C) A rhombus
D) A cyclic quadrilateral
Explanation
(b)
$\mathcal{z}_{1} - \mathcal{z}_{4} = \mathcal{z}_{2} - \mathcal{z}_{3}$
$\Rightarrow \ \frac{\mathcal{z}_{1} + \mathcal{z}_{3}}{2} = \frac{\mathcal{z}_{2} + \mathcal{z}_{4}}{2}$
$\Rightarrow \$Diagonals bisect each other
Given that, $\arg{\left( \frac{\mathcal{z}_{4} - \mathcal{z}_{1}}{\mathcal{z}_{2} - \mathcal{z}_{1}} \right) = \frac{\pi}{2}}$
$\Rightarrow$ Angle at $\mathcal{z}_{1} = \frac{\pi}{2}$
So, it form a rectangle
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.