In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. If $BM = \sqrt{30}$ and $CM = 3$,find $AC$.

$A$ line drawn parallel to $\overline{YZ}$ in the plane of $\Delta XYZ$ passes through the midpoint of $\overline{XY}$. Prove that this line bisects $\overline{XZ}$.

$E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\Delta PQR$. For the given case,state whether $EF || QR$:
$PE = 4 \, cm, QE = 4.5 \, cm, PF = 8 \, cm$ and $RF = 9 \, cm$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 2x$,$BM = 3x + 5$,and $CM = 8x$,find the value of $x$.

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

In $\Delta PQR$,$P-X-Q$,$P-Y-R$ and $\overline{XY} \parallel \overline{QR}$. If $PY = 4$,$YR = 7$ and $XY = 11$,find $QR$.