First slide

Geometry of complex numbers

Question

$z_{1}, z_{2}, z_{3}, z_{4}$ are distinct complex numbers representing the vertices of a quadrilateral $A B C D$ taken in order. If $z_{1} - z_{4} = z_{2} - z_{3}$ and $\arg (z_{4} - z_{1}) / (z_{2} - z_{1}) = π / 2,$ then the quadrilateral is

Very difficult

A

rectangle
B

rhombus
C

square
D

trapezium

Solution

The first condition implies that $[(z_{1} + z_{3}) / 2] = [(z_{2} + z_{4}) / 2],$ diagonals $A C$ and $B D$ bisect each other.
Hence, quadrilateral is a parallelogram.
The second condition implies that the angle between $A D$ and $A B$ is $90^{\circ}$ .
Hence the parallelogram is a rectangle.

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