Question

If $z_1,z_2,z_3$ are vertices of an equilateral triangle $ABC$ such that $|z_1-i|=|z_2-i|=|z_3-i|$, then $|z_1+z_2+z_3|$ equals to

• A. $3\sqrt{3}$
• B. $\sqrt{3}$
• C. $3$
• D. $\frac{1}{3\sqrt{3}}$

Answer 1

The correct option is C $3$

Given that,
$|z_1-i|=|z_2-i|=|z_3-i|$
Hence, $z_1,z_2,z_3$ lie on a circle whose centre is $i$.
Also, given that the triangle is equilateral. Hence, centroid and circumcenter coincides.
$\therefore \frac{z_1+z_2+z_3}{3}=i$
$\Rightarrow |z_1+z_2+z_3|=3$