In rhombus $PQRS$,$PR = 40 \, cm$ and $QS = 42 \, cm$,then $PQ = \dots \dots \dots \, cm$.

State whether each of the following statements is true or false:
$(1)$ $ABCD$ is a rhombus. If $AB = 12 \, \text{cm}$, then the perimeter of $ABCD$ is $48 \, \text{cm}$.
$(2)$ $PQRS$ is a square. If $PR = 8 \, \text{cm}$, then the perimeter of $PQRS$ is $32 \, \text{cm}$.

$ABCD$ is a rectangle. If $AB = 12 \, cm$ and $BC = 7 \, cm$,then find the area of $ABCD$ in $cm^2$.

$P$ and $Q$ are points on opposite sides $AD$ and $BC$ of a parallelogram $ABCD$ such that $PQ$ passes through the point of intersection $O$ of its diagonals $AC$ and $BD$. Show that $PQ$ is bisected at $O$.

$AB$ and $CD$ are respectively the smallest and longest sides of a quadrilateral $ABCD$ (see Fig). Show that $\angle A > \angle C$ and $\angle B > \angle D$.

$E$ and $F$ are points on diagonal $AC$ of a parallelogram $ABCD$ such that $AE = CF$. Show that $BFDE$ is a parallelogram.