The quadrilateral formed by joining the mid-points of the sides of a quadrilateral $PQRS,$ taken in order,is a rhombus,if

$PQ$ and $RS$ are two equal and parallel line segments. Any point $M$ not lying on $PQ$ or $RS$ is joined to $Q$ and $S$. Lines are drawn through $P$ parallel to $QM$ and through $R$ parallel to $SM$,meeting at $N$. Prove that line segments $MN$ and $PQ$ are equal and parallel to each other.

State whether each of the following statements is true or false:
$(1)$ $ABCD$ is a parallelogram. If $AB = 12 \text{ cm}$ and $BC = 5 \text{ cm}$,then $AC = 13 \text{ cm}$.
$(2)$ $ABCD$ is a trapezium. If $AB \parallel CD$ and $AB = 10 \text{ cm}$,then $CD = 10 \text{ cm}$.

The perimeter of rectangle $ABCD$ is $112 \, cm$ and $AB : BC = 5 : 3$. Find the length of $AB$ in $cm$.

In $\Delta PQR$,$A$,$B$,and $C$ are the mid-points of $PQ$,$QR$,and $RP$ respectively. If the perimeter of $\Delta ABC$ is $18.6 \, cm$,then the perimeter of $\Delta PQR$ is $\ldots \ldots \ldots cm$.

In $\Delta ABC$,$P$ and $Q$ are the midpoints of $AB$ and $AC$ respectively. If $PQ + BC = 8.4 \, cm$ and $AB + AC = 20.5 \, cm$,then find the perimeter of $\Delta ABC$ in $cm$.