Question

If $|z_1|=|z_2|=|z_3|$ and $z_1,z_2,z_3$ are vertices of an equilateral triangle, then what is the value of $z_1+z_2+z_3$

• A. $0$
• B. $1$
• C. $3$
• D. none of these

Answer 1

Answer: (A)

$0$

If $A=z_1$, $B=z_2$ and $C=z_3$ are vertices of the triangle, then its centroid G will be $\frac{(z_1+z_2+z_3)}{3}$

Now,$|z_1|=|z_2|=|z_3|=>$ the points $A,B,C$ lie on a circle with centre as the origin $O$.

Therefore, $O$ is the circumcentre of the triangle.

But for an equilateral triangle, the circumcentre coincides with the centroid.

Therefore, the centroid of the triangle is $O$

or

$\frac{(z_1+z_2+z_3)}{3}=0$

or

$(z_1+z_2+z_3)=0$