Question

If three complex numbers $z_1,z_2$ and $z_3$ satisfy $\frac{z_1-z_3}{z_2-z_3}=\frac{1-i\sqrt{3}}{2}$, then the triangle formed by $z_1,z_2$ and $z_3$ is

• A. scalene
• B. right-angled isosceles
• C. equilateral
• D. obtuse-angled isosceles

Answer 1

The correct option is C equilateral

$\frac{z_1-z_3}{z_2-z_3}=\frac{1-i\sqrt{3}}{2}\Rightarrow \frac{z_1-z_3}{z_2-z_3}=e^{-i\pi /3}\cdots \left(1\right)$

Now using the rotation concept we get
$\frac{z_1-z_3}{z_2-z_3}=\frac{|z_1-z_3|}{|z_2-z_3|}{e}^{-i\pi /3}\cdots \left(2\right)$
comparing $\left(1\right)$ and $\left(2\right)$, we get
$|z_1-z_3|=|z_2-z_3|$
and angle between any two sides is $60°$
Hence the given triangle is equilateral triangle.