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Argument of a Complex Number

Let z1, z2, z...

Question

Let $z_{1}, z_{2}, z_{3}, z_{4}$ are distinct complex numbers representing the vertices of a quadrilateral $A B C D$ taken in order. If $z_{1} - z_{3} = z_{2} - z_{4}$ and $arg (\frac{z_{4} - z_{1}}{z_{2} - z_{1}}) = \frac{π}{2},$ then the quadrilateral is

A

rectangle

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B

rhombus

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C

square

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D

trapezium

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Solution

The correct option is A rectangle

From $z_{1} - z_{2} = z_{3} - z_{4} \Rightarrow | z_{1} - z_{2} | = | z_{3} - z_{4} |$
and $arg (\frac{z_{1} - z_{2}}{z_{3} - z_{4}}) = 0$
$∴ A B = C D, A B ∥ C D$
Now $arg (\frac{z_{4} - z_{1}}{z_{2} - z_{1}}) = \frac{π}{2}$
$\Rightarrow A D ⊥ A B$
Hence, $A B C D$ will be rectangle.

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Argument of a Complex Number

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