Question

A triangle is formed on the Argand plane by the complex numbers $z,iz$ and $z+iz$. If $|z|=2$, then area of triangle is sq. unit

Answer 1

Let the vertices of the triangle be $A\left(z\right),B\left(iz\right),C(z+iz)$.
We know that $iz$ is obtained by rotating $OA$ through an angle ${90}^{\circ }$.
From the diagram, $OA=OB=BC=AC,\angle O={90}^{\circ }$
Hence $OABC$ will be a square

From Argand diagram, it is clear that
Area of $△ABC$ = Area of $△OAB$
$=\frac{1}{2}\times OA\times \left(OB\right)[\because \text{it is right angled at point O}]$
$=\frac{1}{2}|z|\times |iz|$
$=\frac{1}{2}|z{|}^2=2$ sq. unit