In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QM}$ is an altitude. If $QM = 6$ and $MR = 9$,then $PR = \ldots$

The lengths of the sides of a triangle are $m^{2}+n^{2}$,$2mn$,and $m^{2}-n^{2}$,where $m > n > 0$. Prove that the triangle is a right-angled triangle.

In $\Delta XYZ,$ the bisector of $\angle X$ intersects $\overline{YZ}$ at $T$. If $XY = 10, XZ = 14$ and $YT = 5,$ find $YZ.$

$O$ is any point inside a rectangle $ABCD$ (see Figure). Prove that $OB^{2} + OD^{2} = OA^{2} + OC^{2}$.

$ABCD$ is a trapezium in which $AB \parallel DC$ and $P$ and $Q$ are points on $AD$ and $BC$ respectively such that $PQ \parallel DC$. If $PD = 18 \, cm$,$BQ = 35 \, cm$ and $QC = 15 \, cm$,find $AD$ (in $cm$).

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = 2$,$MB = 5$ and $MN = 4$,find $BC$.