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

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 8$ and $CM = 2$,then $BM = \ldots$

In $\Delta ABC$,$B-D-C$. The bisectors of $\angle ADB$ and $\angle ADC$ intersect $\overline{AB}$ and $\overline{AC}$ at $P$ and $Q$ respectively. Prove that $AP \times AQ \times BD \times DC = AD^2 \times PB \times QC$. From this,prove that $D$ is the midpoint of $\overline{BC}$ if $\overleftrightarrow{PQ} \parallel \overleftrightarrow{BC}$.

$A$ $6.5 \, m$ long ladder leans on a wall to reach a height of $6 \, m$ on the wall. Then,the lower end of the ladder remains $\ldots \ldots \ldots \ldots \, m$ away from the wall.

In $\Delta XYZ$,$\overline{XN}$ is a median. If $XY^2 + XZ^2 = 200$ and $XN = 8$,find $YZ$.

If $S$ is a point on side $PQ$ of a $\triangle PQR$ such that $PS = QS = RS$,then