The area of the triangle with vertices P ( z ) , Q ( iz ) and R ( z...

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.,

$A = \dfrac{1}{2} \times base \times height$

$⇒ A = \dfrac{1}{2} \times \left|z\right| \times \left|iz\right|$

$⇒ A = \dfrac{1}{2}{\left|z\right|}^{2}$

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