Show that the area of the triangle on the Argand diagram formed by the complex number `z ,i za n dz+i z` is `1/2|z|^2`

Explanation

Let the vertices of the triangle be `A(z),B(iz),C(z+iz)`. We know that iz is obtained by rotating OA through an angle `90^(@)` in anticlockwise direction. Also, point `z+iz` can be obtained by completing the parallelogram two of whose adjacent sides are OA and OB. From Argand diagram, it is clear that

Area of `DeltaABC=" Area of "DeltaOAB`
`=(1)/(2)xxOAxxOB" "[because" it is right angled at point O"]`
`=(1)/(2)|z|xx|iz|`
`=(1)/(2)|z|^(2)`