$ABCD$ is a square. If $AC = 16 \, cm$,then find the area of $ABCD$ in $cm^2$.

$PQRS$ is a rectangle. If $PQ = 20 \, cm$ and $\operatorname{ar}(PQRS) = 300 \, cm^2$,then $SP = \dots \, cm$.

$ABCD$ is a trapezium with parallel sides $AB = a \text{ cm}$ and $DC = b \text{ cm}$. $E$ and $F$ are the mid-points of the non-parallel sides. The ratio of $\operatorname{ar}(ABFE)$ and $\operatorname{ar}(EFCD)$ is

In $\triangle ABC,$ if $L$ and $M$ are the points on $AB$ and $AC,$ respectively such that $LM \parallel BC.$ Prove that $\operatorname{ar}(\triangle LOB) = \operatorname{ar}(\triangle MOC).$

$X$ and $Y$ are points on the side $LN$ of the triangle $LMN$ such that $LX = XY = YN$. Through $X$,a line is drawn parallel to $LM$ to meet $MN$ at $Z$ (See figure). Prove that $\operatorname{ar}(LZY) = \operatorname{ar}(MZYX)$.

The diagonals of a parallelogram $ABCD$ intersect at a point $O$. Through $O$,a line is drawn to intersect $AD$ at $P$ and $BC$ at $Q$. Show that $PQ$ divides the parallelogram into two parts of equal area.