If the four complex numbers z,-z,-z-2Re(-z) and z-2Re(z) represent the vertices of a squar…

Question

If the four complex numbers $z,-z,-z-2\operatorname{Re}(-z)$ and $z-2\operatorname{Re}(z)$ represent the vertices of a square of side 4 units in the Argand plane, then $|z|$ is equal to:

Accepted Answer

Let $z=x+iy$. Side length: $|z-\bar z|=4 \implies |2iy|=4 \implies |y|=2$. Also $|z-(\bar z-2\operatorname{Re}(z))|=4 \implies |2\operatorname{Re}(z)|=4 \implies |x|=2$.

$$|z|=\sqrt{x^2+y^2}=\sqrt{4+4}=2\sqrt2.$$

Solutions · 1