Question

The points $z_1,z_2,z_3,z_4$ in the complex plane are the vertices of a parallelogram taken in order if and only if

• A. $z_1+z_4=z_2+z_3$
• B. $z_1+z_3=z_2+z_4$
• C. $z_1+z_2=z_3+z_4$
• D. $Noneofthese$

Answer 1

The correct option is B

$z_1+z_3=z_2+z_4$

Explanation for the correct option:

Given, $z_1,z_2,z_3,z_4$ are the vertices of a parallelogram.

Let $PQRS$ be the parallelogram,

In a parallelogram diagonals bisect each other.

Therefore, mid-point of $PR$ and $QS$ is same.

$\Rightarrow \frac{z_1+z_3}{2}=\frac{z_2+z_4}{2}$

$\Rightarrow {\mathit{z}}_{\mathbf{1}}\mathbf{}\mathbf{+}{\mathit{z}}_{\mathbf{3}}\mathbf{}\mathbf{=}{\mathit{z}}_{\mathbf{2}\mathbf{}}\mathbf{+}{\mathit{z}}_{\mathbf{4}}\mathbf{}$

Hence, Option ‘B’ is Correct.