The complex numbers z_1, z_2 and z_3 satisfyi | vedclass

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(C) Given $\frac{z_1 - z_3}{z_2 - z_3} = \frac{1}{2} - i\frac{\sqrt{3}}{2}$.Taking the modulus on both sides,we get $\left| \frac{z_1 - z_3}{z_2 - z_3

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Equilateral

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(C) Given $\frac{z_1 - z_3}{z_2 - z_3} = \frac{1}{2} - i\frac{\sqrt{3}}{2}$.Taking the modulus on both sides,we get $\left| \frac{z_1 - z_3}{z_2 - z_3} ight| = \sqrt{\left( \frac{1}{2} ight)^2 + \left( -\frac{\sqrt{3}}{2} ight)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = 1$.This implies $|z_1 - z_3| = |z_2 - z_3|$,meaning two sides of the triangle are equal.Now,the argument of the complex number is $	ext{amp}\left( \frac{z_1 - z_3}{z_2 - z_3} ight) = 	an^{-1}\left( \frac{-\sqrt{3}/2}{1/2} ight) = 	an^{-1}(-\sqrt{3}) = -\frac{\pi}{3}$.This means the angle between the sides $z_1 - z_3$ and $z_2 - z_3$ is $60^\circ$.Since two sides are equal and the included angle is $60^\circ$,the triangle is equilateral.

The complex numbers $z_1, z_2$ and $z_3$ satisfying $\frac{z_1 - z_3}{z_2 - z_3} = \frac{1 - i\sqrt{3}}{2}$ are the vertices of a triangle which is

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