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 $, cm$)

$ABCD$ is a trapezium with $AB || DC$. $A$ line parallel to $AC$ intersects $AB$ at $X$ and $BC$ at $Y$. Prove that $\operatorname{ar}(ADX) = \operatorname{ar}(ACY)$. [Hint: Join $CX$.]

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

In the figure,$ABC$ and $BDE$ are two equilateral triangles such that $D$ is the mid-point of $BC$. If $AE$ intersects $BC$ at $F$,show that:
$(i)$ $\operatorname{ar}(BDE) = \frac{1}{4} \operatorname{ar}(ABC)$
$(ii)$ $\operatorname{ar}(BDE) = \frac{1}{2} \operatorname{ar}(BAE)$
$(iii)$ $\operatorname{ar}(ABC) = 2 \operatorname{ar}(BEC)$
$(iv)$ $\operatorname{ar}(BFE) = \operatorname{ar}(AFD)$
$(v)$ $\operatorname{ar}(BFE) = 2 \operatorname{ar}(FED)$
$(vi)$ $\operatorname{ar}(FED) = \frac{1}{8} \operatorname{ar}(AFC)$

$A$ farmer has a field in the form of a parallelogram $PQRS$. She took any point $A$ on $RS$ and joined it to points $P$ and $Q$. In how many parts is the field divided? What are the shapes of these parts? The farmer wants to sow wheat and pulses in equal portions of the field separately. How should she do it?

In the figure,diagonals $AC$ and $BD$ of quadrilateral $ABCD$ intersect at $O$ such that $OB = OD$. If $AB = CD$,then show that:
$(i)$ $ar(DOC) = ar(AOB)$
$(ii)$ $ar(DCB) = ar(ACB)$
$(iii)$ $DA \parallel CB$ or $ABCD$ is a parallelogram.
[Hint: From $D$ and $B$,draw perpendiculars to $AC$.]