Let z 1- z 2- z 3- z 4 are distinct complex numbers representing the vertices of a quadrilateral ABCD taken in order- If z 1- z 3- z 2- z 4 and z 4- z 1- z 2- z 1-2- then the quadrilateral isA- rhombusB- squareC- rectangleD- trapezium

Given: nbsp; nbsp;z_1 z_3 = z_2 z_4 , (z_4 z_1z_2 z_1)=π2 From z_1 z_2=z_3 z_4 ⇒ |z_1 z_2|=|z_3 z_4| and(z_1 z_2z_3 z_4)=0 ∴ AB=CD,AB∥ CD Now, ...

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Byju's Answer

Standard XII

Mathematics

Properties of Cube Root of a Complex Number

Let z 1, z 2,...

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

rectangle

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rhombus

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square

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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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