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 $cm$.

In $\Delta ABC$,$X$ and $Y$ are the midpoints of $AB$ and $AC$ respectively. State the type of quadrilateral $XBCY$.

$ABC$ 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 $MD \perp AC$.

$D$ and $E$ are the mid-points of the sides $AB$ and $AC$ of $\Delta ABC$ and $O$ is any point on side $BC$. $O$ is joined to $A$. If $P$ and $Q$ are the mid-points of $OB$ and $OC$ respectively,then $DEQP$ is

Prove that the line segment joining the mid-points of the non-parallel sides of a trapezium is parallel to the parallel sides of the trapezium.

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 $\text{ cm}$)