Question
The points $$z_1, z_2 , z_3, z_4$$ in a complex plane are vertices of a parallelogram taken in order, then
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.
None of these
Answer :
$${z_1} + {z_3} = {z_2} + {z_4}$$
Solution :
Let $$z_1 , z_2 , z_3$$ and $$z_4$$ the points in complex plane be the vertices of a parallelogram taken in order.
Since, the diagonals of a parallelogram bisect,
hence, the mid points of $$AC$$ and $$BD$$ must coincide
i.e.,
$$\eqalign{ & \frac{{{z_1} + {z_3}}}{2} = \frac{{{z_2} + {z_4}}}{2} \cr & \Rightarrow {z_1} + {z_3} = {z_2} + {z_4} \cr} $$
Let $$z_1 , z_2 , z_3$$ and $$z_4$$ the points in complex plane be the vertices of a parallelogram taken in order.
Since, the diagonals of a parallelogram bisect,
hence, the mid points of $$AC$$ and $$BD$$ must coincide
i.e.,
$$\eqalign{ & \frac{{{z_1} + {z_3}}}{2} = \frac{{{z_2} + {z_4}}}{2} \cr & \Rightarrow {z_1} + {z_3} = {z_2} + {z_4} \cr} $$