Find condition for which `z_(1),z_(2),z_(3)` represent vertices of an equilateral triangle .

Options

A. `z_(1)+z_(2)=z_(3)`
B. `z_(2)+z_(3)=z_(1)`
C. `z_(1)+z_(3)=z_(2)`
D. `z_(1)+z_(2)+z_(3)=0`

Answer

D

Explanation

Let ABC be a triangle such that the complex numbers, `z_(1),z_(2)` and `z_(3)` represent the vertices A,B and C respectively.
Now, `|z_(1)|=|z_(2)|=|z_(3)|`
`rArr |z_(1)-0|=|z_(2)-0|=|z_(3)-0|`
`rArr` OA=OB=OC , where O is the origin.
`rArr` Origin is the circumcenter of triangle ABC.
But, the circumcenter of an equilateral triangle is same as its centroid, So, the centroid of triangle ABC is the origin.
`rArr (z_(1)+z_(2)+z_(3))/3=0` [`therefore` Affix of the centroid `=(z_(1)+z_(2)+z_(3))/(3)`]
`rArr z_(1)+z_(2)+z_(3)=0`