Question

If $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$, then area of the triangle whose vertices are $z_1,z_2,z_3$ is

• A. $\frac{3\sqrt{3}}{4}\text{sq. unit}$
• B. $\frac{\sqrt{3}}{4}\text{sq. units}$
• C. $1\text{sq. unit}$
• D. $2\text{sq. units}$

Answer 1

The correct option is A $\frac{3\sqrt{3}}{4}\text{sq. unit}$

Centroid $=\frac{z_1+z_2+z_3}{3}=0$
As $|z_1|=|z_2|=|z_3|=1$
So, distance from origin and vertices are same,
Circumcentre is also origin
Therefore, the triangle is equilateral.
Let the side length is $a$


From diagram, we get
$\frac{a}{2}=1\cos {30}^{\circ }\Rightarrow a=\sqrt{3}\text{units}$
Hence, then area of equilateral triangle
$=\frac{\sqrt{3}}{4}{a}^2=\frac{3\sqrt{3}}{4}\text{sq. units}$