The points z1, z2, z3, z4 in the complex plane are the vertices of a parallelogram taken in order, iff :

Question

The points z1, z2, z3, z4 in the complex plane are the vertices of a parallelogram taken in order, iff : (A) z1+z4=z2+z3 (B) z1+z3=z2+z4 (C) z1+z2=z3+z4 (D) none of these

Tardigrade Admin · Accepted Answer

Diagonals of a parallelogram bisect each other at the same point.
Since, the points

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

and

z_{4}

are the vertices of a parallelogram, therefore diagonals of a parallelogram bisect each other at a point

\frac{z _{1} + z _{3}}{2} = \frac{z _{2} + z _{4}}{2}

\Rightarrow z_{1} + z_{3} = z_{2} + z_{4}

Q. The points $z_{1}, z_{2}, z_{3}, z_{4}$ in the complex plane are the vertices of a parallelogram taken in order, iff :

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

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

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

none of these

Q. The points z1​,z2​,z3​,z4​ in the complex plane are the vertices of a parallelogram taken in order, iff :

z1​+z4​=z2​+z3​

z1​+z3​=z2​+z4​

z1​+z2​=z3​+z4​

none of these

Diagonals of a parallelogram bisect each other at the same point. Since, the points z1​,z2​,z3​ and z4​ are the vertices of a parallelogram, therefore diagonals of a parallelogram bisect each other at a point 2z1​+z3​​=2z2​+z4​​ ⇒z1​+z3​=z2​+z4​

Q. The points $z_{1}, z_{2}, z_{3}, z_{4}$ in the complex plane are the vertices of a parallelogram taken in order, iff :

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

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

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