Question

Le $A=\frac{2}{3}{e}^{\frac{i\pi }{2}},B=\frac{2}{3}{e}^{\frac{i\pi }{6}},C=\frac{2}{3}{e}^{\frac{i5\pi }{6}}$ be the three points forming a $△ABC$ in the argand, then $△ABC$ is:

• A. scalene triangle
• B. an equilateral triangle
• C. an isosceles triangle
• D. N.O.T

Answer 1

Answer: (C)

an isosceles triangle

From the given information, we can calculate the measure of angles $A,B,C$

Let $A=z_1;B=z_2;C=z_3$, then

$\angle A=\frac{z_2-z_1}{z_3-z_1}=\frac{\frac{2}{3}\angle 30-\frac{2}{3}\angle 90}{\frac{2}{3}\angle 150-\frac{2}{3}\angle 90}=\frac{\frac{2}{3}\angle -30}{\frac{2}{3}\angle -150}={120}^0$

$\angle B=\frac{z_3-z_2}{z_1-z_2}=\frac{\frac{2}{3}\angle 150-\frac{2}{3}\angle 30}{\frac{2}{3}\angle 90-\frac{2}{3}\angle 30}=\frac{\frac{2}{3}\angle 180}{\frac{2}{3}\angle 150}={30}^0$

$\angle C=\frac{z_1-z_3}{z_2-z_3}=\frac{\frac{2}{3}\angle 90-\frac{2}{3}\angle 150}{\frac{2}{3}\angle 30-\frac{2}{3}\angle 150}=\frac{\frac{2}{3}\angle 30}{\frac{2}{3}\angle 0}={30}^0$

An isosceles triangle is a triangle with (at least) two equal sides.

Here we can see that $\angle B=\angle C$

Hence $△ABC$ is isosceles