In $\Delta PQR$,$PM$ is a median. If $\text{ar}(\Delta PMQ) = 36 \text{ cm}^2$,then find $\text{ar}(\Delta PMR)$ in $\text{cm}^2$.

$X$ 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 perimeter of square $ABCD$ is $16 \, cm$,then $ar(ABCD) = \ldots \ldots \ldots \, cm^2$.

In parallelogram $PQRS$,$PQ = 15 \, cm$. Altitudes $SM$ and $SN$ are corresponding to bases $PQ$ and $QR$ respectively. If $SM = 6 \, cm$ and $SN = 10 \, cm$,find $QR$ and the perimeter of $PQRS$.

In $\Delta ABC$,$\angle B = 90^{\circ}$,$BC = 8 \, \text{cm}$,and $AC = 17 \, \text{cm}$. $BE$ is a median of the triangle and $M$ is the midpoint of $BE$. Find the area of $\Delta BMC$ in $\text{cm}^2$.

In parallelogram $ABCD$,$AB = 20 \, cm$. Altitudes $AY$ and $DX$ are corresponding to bases $BC$ and $AB$ respectively. If $DX = 12 \, cm$ and $AY = 15 \, cm$,then find $BC$ and the perimeter of $ABCD$.