In $\Delta ABC$,$P$,$Q$,and $R$ are the midpoints of the sides. If the perimeter of $\Delta ABC$ is $23 \, cm$,then find the perimeter of $\Delta PQR$ in $cm$.

$ABCD$ is a quadrilateral in which $P$,$Q$,$R$ and $S$ are mid-points of the sides $AB$,$BC$,$CD$ and $DA$ respectively (see figure). $AC$ is a diagonal. Show that: $PQ = SR$.

In $\Delta ABC$,$P$ and $Q$ are the midpoints of $AB$ and $AC$ respectively. If $BC + PQ = 21 \text{ cm}$,then find the length of $BC$.

$ABCD$ is a quadrilateral in which $P$,$Q$,$R$ and $S$ are mid-points of the sides $AB$,$BC$,$CD$ and $DA$ (see Fig). $AC$ is a diagonal. Show that: $PQRS$ is a parallelogram.

$\Delta ABC$ is a triangle right-angled at $C$. $A$ line through the mid-point $M$ of hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that $D$ is the mid-point of $AC$.

In $\Delta ABC$, $P$, $Q$, and $R$ are the midpoints of $AB$, $BC$, and $CA$ respectively. If $AB = 8.2 \text{ cm}$, $BC = 6.4 \text{ cm}$, and $CA = 7.5 \text{ cm}$, then find the perimeter of quadrilateral $PBQR$. (in $\text{ cm}$)