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Byju's Answer

Standard XII

Mathematics

Combination

z1, z2, z3, z...

Question

$z_{1}, z_{2}, z_{3}, z_{4}$ are distinct complex numbers representing the vertices of a quadrilateral ABCD taken in order. If $z_{1} - z_{4} = z_{2} - z_{3}$ and $a r g [(z_{4} - z_{1}) / (z_{2} - z_{1})] = \frac{π}{2}$ , 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 D Rectangle
Consider
$A = z_{1}$
$B = z_{2}$
$C = z_{3}$
$D = z_{4}$

Now

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

Therefore

$A D | | B C$ and $A D = B C$

Therefore, the opposite sides are equal and parallel.
Hence it is a quadrilateral.

Now
$a r g \frac{z_{4} - z_{1}}{z_{2} - z_{3}} = \frac{π}{2}$

Hence $∠ A = \frac{π}{2}$

Hence $∠ A = ∠ B = ∠ C = ∠ D = 90^{0}$

Therefore, it is a rectangle.

Hence, option 'A' is correct.

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