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Q.
The centre of a square $ABCD $ is at $ z = 0$. The affix of the vertex $A$ is ${z_1}$. Then the affix of the centroid of the triangle $ABC$ is
Complex Numbers and Quadratic Equations
Solution:
Affix of $A$ is $z_1$ means $OA = z_1$ and $OB$ and $OC$ are obtained by rotating $OA$ thro $\frac{\pi}{2}$ and $\pi$.
$\therefore $ Affix of $B$ and $C$ are $i\,z_1$ and $- z_i$ resp.
Hence the Affix of the centroid of $\Delta ABC$ is
$\frac{z_{1}+iz_{1}+\left(-z_{1}\right)}{3}=\frac{iz_{1}}{3}$.
$=\frac{1}{3}z_{1}\left[cos \frac{\pi}{2}+i\,sin \frac{\pi}{2}\right]$
If $A$, $B$, $C$ are taken in clockwise sense, then the Affix of the centroid is $\frac{1}{3} z_{1}\left(cos \frac{\pi}{2}-i\,sin \frac{\pi}{2}\right)$.
Thus, the Affix of the centroid is
$\frac{1}{3}z_{1}\left(cos \frac{\pi}{2} \pm i\,sin \frac{\pi}{2}\right)$.
