In the figure,$ABCD$ is a parallelogram,$AE \perp DC$ and $CF \perp AD$. If $AB = 16 \, cm, AE = 8 \, cm$ and $CF = 10 \, cm$,find $AD$. (in $, cm$)

The side $AB$ of a parallelogram $ABCD$ is produced to any point $P$. $A$ line through $A$ and parallel to $CP$ meets $CB$ produced at $Q$ and then parallelogram $PBQR$ is completed. Show that $\text{ar}(ABCD) = \text{ar}(PBQR)$.
[Hint: Join $AC$ and $PQ$. Now compare $\text{ar}(ACQ)$ and $\text{ar}(APQ)$.]

$D, E$ and $F$ are respectively the mid-points of the sides $BC, CA$ and $AB$ of a $\Delta ABC$. Show that $BDEF$ is a parallelogram.

In a triangle $ABC$,$E$ is the mid-point of median $AD$. Show that $\operatorname{ar}(BED) = 1/4 \operatorname{ar}(ABC)$.

Diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB || DC$ intersect each other at $O$. Prove that $ar(AOD) = ar(BOC).$

In the figure,$E$ is any point on the median $AD$ of a $\Delta ABC$. Show that $\text{ar} (ABE) = \text{ar} (ACE)$.