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

Question

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 (A) rhombus (B) square (C) rectangle (D) trapezium

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/84729d2651fabfb0973d707578febdb6-.png" /> Since z1 - z4 = z2 - z3 ⇒ (z1 + z3/2) = (z1 + z4/2) i.e., diagonal AC and DB bisect each other. ∴ ABCD is a parallelogram. Again angle BAD = Arg((z4 - z1/z2 - z1)) = π/2 which means parallelogram is a rectangle.

Q. 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(z2​−z1​z4​−z1​​)=2π​, then quadrilateral is a

rhombus

square

rectangle

trapezium

Since z1​−z4​=z2​−z3​ ⇒2z1​+z3​​=2z1​+z4​​ i.e., diagonal AC and DB bisect each other. ∴ABCD is a parallelogram. Again ∠BAD=Arg(z2​−z1​z4​−z1​​)=π/2 which means parallelogram is a rectangle.

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 $a r g (\frac{z _{4} - z _{1}}{z _{2} - z _{1}}) = \frac{π}{2}$ , then quadrilateral is a