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 $\Delta ABC$,$m \angle B = 90^{\circ}$,$AC - BC = 2$ and $BC - AB = 7$. Find the perimeter of $\Delta ABC$.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = x-3$,$MB = 2x-7$,$AN = x+3$ and $NC = 2x+3$,find the value of $x$.

In $\square PQRS$,$\overline{PQ} \parallel \overline{RS}$ and $\overline{PR} \cap \overline{QS} = \{X\}$. If $XP = 2x + 4$,$XQ = 4x - 2$,$XR = x + 1$,and $XS = 4$,find the value of $x$.

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QS}$ is an altitude. If $PS - SR = 10$ and $PQ^2 - QR^2 = 260$,then $PR = \dots$

In $\Delta ABC$,$m\angle B = 90^{\circ}$. $D$ is the midpoint of $\overline{BC}$ and $F$ is the midpoint of $\overline{AB}$. Prove that $AD^{2} + CF^{2} = \frac{5}{4} AC^{2}$.