Question

Area of triangle formed by the complex numbers $z,iz$ and $z+iz$ where $z=x+iy$ is

• A. $|z|$
• B. $2|z{|}^2$
• C. $|z^2|$
• D. $\frac{1}{2}|z{|}^2$

Answer 1

The correct option is D $\frac{1}{2}|z{|}^2$

Given,

$a=z$

$b=iz$

$c=z+iz$

$|ab|=|zi-z|$

$|bc|=|z|$

$|ac|=|iz|$

$|ac|=|bc|$

let $p$ be the midpoint of $ab$

$p=\frac{z+zi}{2}$

$|pc|=\frac{z+zi}{2}$

area of triangle $=\frac{1}{2}|ab\times pc|$

$=\frac{1}{2}\times \frac{z+zi}{2}\times (iz-z)$

$=\frac{1}{4}\times |-2z^2|$

$=\frac{|z{|}^2}{2}$