Prove that of all parallelograms of which the sides are given, the parallelogram which is rectangle has the greatest area.

In Figure, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram whose diagonals $\mathrm{A}\mathrm{C}$ and $\mathrm{B}\mathrm{D}$ intersect at $\mathrm{O}$. A line segment through $\mathrm{O}$ meets $\mathrm{A}\mathrm{B}$ at $\mathrm{P}$ and $\mathrm{D}\mathrm{C}$ at $\mathrm{Q}$. Prove that ar $\left(\mathrm{A}\mathrm{P}\mathrm{Q}\mathrm{D}\right)=1/2\mathrm{a}\mathrm{r}(\parallel \mathrm{g}\mathrm{m}\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D})$

In Figure, diagonals $\mathrm{A}\mathrm{C}$ and $\mathrm{B}\mathrm{D}$ of a quadrileteral $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ intersect at $\mathrm{O}$. If $\mathrm{B}\mathrm{O}=\mathrm{O}\mathrm{D}$, prove that $\mathrm{a}\mathrm{r}\mathrm{}(∆\mathrm{A}\mathrm{B}\mathrm{C})=\mathrm{a}\mathrm{r}\mathrm{}(∆\mathrm{A}\mathrm{D}\mathrm{C})$.

In Figure, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram and $\mathrm{O}$ is any point on diagonal $\mathrm{A}\mathrm{C}$. Show that ar $(∆\mathrm{A}\mathrm{O}\mathrm{B})=\mathrm{a}\mathrm{r}\mathrm{}(∆\mathrm{A}\mathrm{O}\mathrm{D})$.

In Figure, $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a quadrilateral. A line through $\mathrm{D}$ parallel to $\mathrm{A}\mathrm{C}$, meets $\mathrm{B}\mathrm{C}$ produced in $\mathrm{P}$. Prove that $\mathrm{a}\mathrm{r}\mathrm{}(∆\mathrm{A}\mathrm{B}\mathrm{P})=\mathrm{a}\mathrm{r}\mathrm{}\left(\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}\right)$

In Figure, $ABCD$ is a parallelogram with area $36 \mathrm{sq}.\mathrm{units}$ $\mathrm{A}\mathrm{Q}\mathrm{B}$ is a triangle. $\mathrm{QD}$ is the median of the triangle Find $\mathrm{ar}\left(\mathrm{AQD}\right)$. Justify your answer.