A and B are the points representing the complex numbers z and (1 + i )z in an Argand diagram. Prove that triangle OAB is a right-angled isoceles triangle, where O is the origin.
Given:
$\displaystyle OA = z, OB = (1 + i)z $
By the definition of multiplication of complex numbers if $\displaystyle z_1 = r_1. cis {\theta}_1, z_1 = r_2. cis {\theta}_2$ we have
$\displaystyle z_1.z_2 = r_1.r_2 cis({\theta}_1 + {\theta}_2)$ so we have $\displaystyle z_1 = (1 + i) = \sqrt 2 .cis {\frac{\pi}{4}}$.
$\displaystyle \therefore$ in $\displaystyle \Delta OAB, |OB| = \sqrt 2.|OA|, \; \angle AOB = \frac{\pi}{4}$
$\displaystyle \therefore \Delta OAB $ is right isosceles triangle.
Kalyan