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 $\frac{3\sqrt{3}}{4}\text{sq. unit}$

$\because z_1+z_2+z_3=0$
$\therefore$ triangle formed by $z_1,z_2,z_3$ will have centriod as origin
Now $|z_1|=|z_2|=|z_3|=1$
Hence distance of each vertices from origin is equal
$\therefore$ we can say that circumcentre is also origin
Hence triangle will be equilateral triangle ($\because$ circumcentre and centriod are same then orthocentre will also be origin)
Hence side of required triangle will be
$a=\sqrt{3}|z_1|=\sqrt{3}$ unit
Hence area will be
$=\frac{\sqrt{3}}{4}{a}^2=\frac{3\sqrt{3}}{4}$sq. unit