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  4. If z1, z2, z3, z4 are four distinct complex numbers representing the vertices of a quadrilateral taken in order such that z1 - z4 = z2 - z3 and arg((z4 - z1/z2 - z1)) = (π/2), then quadrilateral is a

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Q. If $z_1, z_2, z_3, z_4$ are four distinct complex numbers representing the vertices of a quadrilateral taken in order such that $z_1 - z_4 = z_2 - z_3$ and $arg\left(\frac{z_4 - z_1}{z_2 - z_1}\right) = \frac{\pi}{2}$, then quadrilateral is a

Complex Numbers and Quadratic Equations

Solution:

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Since $z_1 - z_4 = z_2 - z_3$
$\Rightarrow \frac{z_1 + z_3}{2} = \frac{z_1 + z_4}{2}$
i.e., diagonal $AC$ and $DB$ bisect each other.
$\therefore ABCD$ is a parallelogram.
Again $\angle BAD = Arg\left(\frac{z_4 - z_1}{z_2 - z_1}\right) = \pi/2$
which means parallelogram is a rectangle.