The complex numbers z1, z2 and z3 satisfying (z1-z3)/(z2-z3)=(1-i √(3))/(2) are the vertices of

step 1 Here in this problem, we have to choose the correct option and we are given the complex number Z

The complex numbers z1, z2 and z3 satisfying...

The complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ satisfying $\frac{\mathrm{z}_{1}-\mathrm{z}_{3}}{\mathrm{z}_{2}-\mathrm{z}_{3}}=\frac{1-\mathrm{i} \sqrt{3}}{2}$ are the vertices of a triangle which is (a) a right angled isosceles triangle (b) an equilateral triangle (c) an obtuse angled isosceles triangle (d) None of the above

The complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}$ and $\mathrm{z}_{3}$ satisfying $\frac{\mathrm{z}_{1}-\mathrm{z}_{3}}{\mathrm{z}_{2}-\mathrm{z}_{3}}=\frac{1-\mathrm{i} \sqrt{3}}{2}$ are the vertices of a triangle which is
(a) a right angled isosceles triangle
(b) an equilateral triangle
(c) an obtuse angled isosceles triangle
(d) None of the above

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Key Concepts

Complex Numbers as Points in the Plane

Complex numbers can be represented as points in a two-dimensional plane, where the real and imaginary parts correspond to x and y coordinates respectively. This connection allows geometric interpretations of algebraic operations, making it possible to analyze shapes and transformations using complex arithmetic.

Complex Multiplication and Rotation

Multiplication by a complex number, especially one with a modulus of 1, corresponds to a rotation in the complex plane. This property is crucial in understanding how relative positions of points change under complex transformations, leading to insights regarding congruence and symmetry in geometric figures.

Triangle Classification via Complex Ratios

By expressing the relationships between vertices of a triangle as ratios of complex numbers, one can deduce structural properties such as side lengths and angles. When these ratios involve complex numbers with modulus 1, it often implies rotations that maintain the triangle's side lengths, which is a key factor in identifying specific types of triangles like equilateral triangles.

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