Points $A,B,C$ are $z_1,z_2$ and $(1-i)z_1+iz_2$ . Then find nature of triangle $ABC$

Question

The points $A,B,C$ represent the complex number $z_1,z_2$ and $(1-i)z_1+iz_2$ respectively on the complex plane. Then triangle $ABC$ is:

$(A)$ Isosceles but not right angled

$(B)$ Right angled but not isosceles

$(C)$ Isosceles and right angled

$(D)$ None of these

Could someone give slight hint as how to proceed in this question?

Answer 1

Denote the third point by $z_3$. You have $$z_3 - z_1 = -iz_1 + iz_2 = i(z_2 - z_3)$$ and $$z_3 - z_2 = (1-i)z_1 + (i-1) z_2 = (1-i)(z_1 - z_2).$$ Compute the modulus in each equation to compare the distances between points.

Deciding whether a triangle is isosceles or right-angled can be done using just these distances.