Question

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

• A. $z_1(\cos \pi \pm i\sin \pi )$
• B. $\frac{z_1}{3}(\cos \pi \pm i\sin \pi )$
• C. $z_1\left(\cos \frac{\pi }{2}\pm i\sin \frac{\pi }{2}\right)$
• D. $\frac{z_1}{3}\left(\cos \frac{\pi }{2}\pm i\sin \frac{\pi }{2}\right)$

Answer 1

Answer: (D)

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
$\frac{z_1+i{z}_1+\left(-z_1\right)}{3}=\frac{i{z}_1}{3}$.
$=\frac{1}{3}{z}_1\left[cos\frac{\pi }{2}+isin\frac{\pi }{2}\right]$
If $A$, $B$, $C$ are taken in clockwise sense, then the Affix of the centroid is $\frac{1}{3}{z}_1\left(cos\frac{\pi }{2}-isin\frac{\pi }{2}\right)$.
Thus, the Affix of the centroid is
$\frac{1}{3}{z}_1\left(cos\frac{\pi }{2}\pm isin\frac{\pi }{2}\right)$.