If $z_{1}, z_{2}, z_{3}, z_{4}$ are the four complex numbers represented by the vertices of a quadrilateral taken in order such that $z_{1} - z_{4} = z_{2} - z_{3}$ and $a m p \frac{z_{4} - z_{1}}{z_{2} - z_{1}} = \frac{π}{2}$ then the quadrilateral is a

square

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rhombus

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rectangle

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a cyclic quadrilateral

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Solution

The correct options are
C rectangle
D a cyclic quadrilateral
We have $z_{1} - z_{4} = z_{2} - z_{3}$ or $\frac{z_{1} + z_{3}}{2} = \frac{z_{2} + z_{4}}{2}$
i.e, the diagonal bisect each other.
$∴$ It is a parallelogram.
Also, $a m p \frac{z_{4} - z_{1}}{z_{2} - z_{1}} = \frac{π}{2}$
$\Rightarrow$ angle at $z_{1}$ is a right angle.
$∴$ It is a rectangle and hence a cyclic quadrilateral.

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Similar questions

Q. If

z_{1}, z_{2}, z_{3}, z_{4}

are the four complex numbers repersented by the vertices of a quadrilateral taken in order such that

z_{1} - z_{4} = z_{2} - z_{3}

and

a m p \frac{z_{4} - z_{1}}{z_{2} - z_{1}} = \frac{π}{2}

then the quadrilateral is a

If $z_{1}, z_{2}, z_{3}, z_{4}$ are the four complex numbers represented by the vertices of a quadrilateral taken in order such that $z_{1} - z_{2} = z_{3} - z_{4}$ and amp $\frac{z_{4} - z_{1}}{z_{2} - z_{1}} = \frac{π}{2}$ , then the quadrilateral is a