Question

If $z_1$ and $z_2$ are distinct complex numbers such that $\left|z_1\right|=\left|z_2\right|=1$ and $z_1+z_2=1$, then the triangle in the complex plane with $z_1,z_2$ and $-1$ as vertices

• A. must be equiliateral
• B. must be right angeled
• C. must be isosceles, but not necessarily equilateral
• D. must be obtuse angles

Answer 1

The correct option is A must be equiliateral

Let $z_1=a+ib$.

Since, $z_1+z_2=1=|z_1{|}^2=|z_2{|}^2$

then, $z_2=1-a-ib$ and

$a^2+b^2=1$
$(a-1{)}^2+b^2=1$

After solving the above equation we get,

$a=\frac{1}{2},b=\pm \frac{\sqrt{3}}{2}$

Hence,

$z_1=\frac{1+i\sqrt{3}}{2},z_2=\frac{1-i\sqrt{3}}{2}$

Now,

$|z_1+1|=|z_2+1|=|z_1-z_2|=\sqrt{3}$

Thus, statement A is true.