Which of the following condition/s in column-II are satisfied by the quadrilateral formed by Z₁, Z₂, Z₃, Z₄ in order given in column-I.

 

 

$$\begin{gathered} \text{ In parallelogram, the midpoints of diagonals coincide } \\ \frac{z_1+z_3}{2}=\frac{z_2+z_4}{2}\Rightarrow z_1-z_4=z_2-z_3 \end{gathered}$$

$$\begin{gathered} \text{ Also in parallelogram, }AB\parallel CD\text{. Hence, } \\ \arg \left(\frac{z_1-z_{\tilde{2}}}{z_3-z_4}\right)=0\Rightarrow \frac{z_1-z_2}{z_3-z_4}\text{ is purely real } \end{gathered}$$

$\text{ In rectangle, adjacent sides are perpendicular. Hence, }$

$\arg \left(\frac{z_1-z_2}{z_3-z_2}\right)=\frac{\pi }{2}\Rightarrow \frac{z_1-z_2}{z_3-z_2}\text{ is purely imaginary }$

$\text{ Also in rectangle, }AC=BD\Rightarrow \left|z_1-z_3\right|=\left|z_2-z_4\right|$

$\text{ In rhombus, }AC\perp BD\Rightarrow \frac{z_1-z_3}{z_2-z_4}\text{ is purely imaginary }$