Question

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

• A. Right angled
• B. Isosceles
• C. Obtuse angled
• D. Equilateral

Answer 1

The correct option is A Right angled

Consider an unit circle centered at origin.
Then $z_1,z_2,z_3$ are points on its circumference.
Now
$z_1+z_2=0$
Hence
$z_1=-z_2$
Thus $z_2$ is the diametrically opposite point to $z_1$.
Hence
$z_1{z}_2$ is the diameter.
Now since $z_3$ is also a point on the circumference of the circle, it is therefore at a unit distance from the origin.
Now $z_1{z}_2$ being the fixed diameter, and the third vertice being on the circumference ultimately gives infinite right angled triangles with $z_1{z}_2$ as hypotenuse and right angled at $z_3$