In $\Delta XYZ$,$m\angle Y = 90^{\circ}$ and $\overline{YP}$ is a median. If $YP = 6$,then $XZ = \ldots$

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. Prove that $\frac{AB^2}{BC^2} = \frac{AM}{CM}$.

In $\Delta MNP$,$\overline{MX}$ is a median. If $MN^2 + MP^2 = 50$ and $MX = 3$,then $NP = \ldots$

In $\Delta PQR$,$m \angle Q = 90^{\circ}$. If $PQ = 33$ and $PR = 65$,find $QR$.

$D$ and $E$ are points on the sides $AB$ and $AC$ of a triangle $ABC$ respectively,such that $AD = 2 \, cm$,$BD = 3 \, cm$,$BC = 7.5 \, cm$,and $DE \parallel BC$. Find the length of $DE$ (in $cm$).

In $\Delta ABC$,$A-P-B$,$A-Q-C$ and $\overleftrightarrow{PQ} \parallel \overleftrightarrow{BC}$. If $AP = 3$,$PB = 5$ and $AQ = 6$,find $QC$ and $AC$.