Question

The quadrilateral formed by the points with affix $0,z,z+iz$ and $iz$ is

• A. Parallelogram
• B. Rhombus
• C. Rectangle
• D. Square

Answer 1

Answer: (D)

Square

Let $z=x+iy$

$iz=-y+ix$

$z+iz=(x+iy)+(-y+ix)$

$=(x-y)+i(x+y)$

Hence the co-ordinates of the quadrilateral are

$A=(0,0)$
$B=(x,y)$

$C=(x-y,x+y)$

$D=(-y,x)$

Therefore

$AB=\sqrt{x^2+y^2}$

$BC=\sqrt{(x-y-x{)}^2+(x+y-x{)}^2}=\sqrt{x^2+y^2}$

$CD=\sqrt{x^2+y^2}=AD$

Hence

$AB=BC=CD=AD$

Therefore all the sides are equal. Hence it is a parallelogram, which is either a square of a rhombus.

$\overrightarrow{AB}=xi+yj$

$\overrightarrow{BC}=-yi+xj$

$\overrightarrow{AB}.\overrightarrow{BC}=-xy+xy=0$ ....(dot product is 0).

Hence

Adjacent sides are perpendicular to one another.

Thus we get a parallelogram with all the sides equal and adjacent sides perpendicular.

Hence it is a square.