Question

Show that the area of the triangle on the Argand diagram formed by the complex numbers $z,iz$ and $z+iz$ is $\frac{1}{2}|z{|}^2$

Answer 1

$z,zi$ have equal magnitude with ${90}^{\circ }$ argument difference.

$\therefore z,zi$ are the remaining $2$ sides, with hypotenuse being $z+iz$ forming a right angled triangle.

Area of triangle $=|\frac{1}{2}\times z\times zi|$

$=\frac{1}{2}|z^2|$

Hence, proved.