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 $BC$.

In $\Delta XYZ$,$A, B$ and $C$ are the midpoints of $XY, YZ$ and $XZ$ respectively. If $AB = 5.2\, cm$,$BC = 4.3\, cm$ and $AC = 6.5\, cm$,then find the perimeter of $\Delta XYZ$ in $cm$.

The figure formed by joining the mid-points of the sides of a quadrilateral $ABCD,$ taken in order,is a square only if,

$D, E$ and $F$ are respectively the mid-points of the sides $BC, CA$ and $AB$ of a $\Delta ABC$. Show that $BDEF$ is a parallelogram.

$ABCD$ is a trapezium in which $AB \parallel DC$,$BD$ is a diagonal,and $E$ is the mid-point of $AD$. $A$ line is drawn through $E$ parallel to $AB$ intersecting $BC$ at $F$. Show that $F$ is the mid-point of $BC$.

In $\triangle ABC$,$AB = 5\, cm$,$BC = 8\, cm$,and $CA = 7\, cm$. If $D$ and $E$ are respectively the mid-points of $AB$ and $BC$,determine the length of $DE$. (in $, cm$)