Question

The area of the triangle whose vertices are complex numbers z, iz, z + iz in the Argand diagram is

• A. $2|z{|}^2$
• B. $1/2|z{|}^2$
• C. $4|z{|}^2$
• D. $|z{|}^2$

Answer 1

Answer: (B)

Vertices of triangle in complex form is $z,iz,z+iz$
In cartesian form vertices are
(x, y), (-y, x) and $(x-y,x+y)$
$\therefore$ Area of triangle $=\frac{1}{2}\left|\begin{array}{ccc}x& y& 1\\ -y& x& 1\\ x-y& x+y& 1\end{array}\right|$
$=\frac{1}{2}\left[x\left(x-x-y\right)-y\left(-y-x+y\right)+1\right]$
$\left(-yx-y^2-x^2+xy\right)$
$=\frac{1}{2}\left[-xy+xy-y^2-x^2\right]=\frac{1}{2}\left(x^2+y^2\right)$
($\because$ Area can not be negative)
$=\frac{1}{2}{\left|z\right|}^2\left(\because z=x+iy,{\left|z\right|}^2=x^2+y^2\right)$