The area of the triangle with vertices P ( z ) ,   Q ( i z ) and R ( z + i z ) is

From the above diagram, it is clear that $z, iz$ are perpendicular to each other.

Here, base $=\left|z\right|$, height $=\left|iz\right|$

So, area of the triangle formed by the vertices i.e., 

$A=\frac{1}{2}\times \mathrm{base}\times \mathrm{height}$

$\Rightarrow A=\frac{1}{2}\times \left|z\right|\times \left|iz\right|$

$\Rightarrow A=\frac{1}{2}{\left|z\right|}^2$

Hence, the required area is $\frac{1}{2}{\left|z\right|}^2$.