Question

$z_1,z_2,z_3,z_4$ are distinct complex numbers representing the vertices of a quadrilateral ABCD taken in order. If $z_1-z_4=z_2-z_3$ and $arg\left[\right(z_4-z_1\left)/\right(z_2-z_1\left)\right]=\frac{\pi }{2}$, then the quadrilateral is

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

Answer 1

Answer: (A)

Rectangle

Consider
$A=z_1$
$B=z_2$
$C=z_3$
$D=z_4$

Now

$z_1-z_4=z_2-z_3$

Therefore

$AD||BC$ and $AD=BC$

Therefore, the opposite sides are equal and parallel.
Hence it is a quadrilateral.

Now
$arg\frac{z_4-z_1}{z_2-z_3}=\frac{\pi }{2}$

Hence $\angle A=\frac{\pi }{2}$

Hence $\angle A=\angle B=\angle C=\angle D={90}^0$

Therefore, it is a rectangle.

Hence, option 'A' is correct.