In $\Delta PQR$,$m\angle P = 90^{\circ}$ and $\overline{PM}$ is an altitude. If $PQ = \sqrt{20}$ and $QM = 4$,then $RM = \ldots \ldots .$

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM : AB = 2 : 3$ and $AC = 15$,then $NC = \ldots$

In $\Delta ABC$,$\overline{AD}$ is a median. If $AB^2 + AC^2 = 338$ and $AD = 5$,find $BC$.

If $\Delta PQR \sim \Delta XYZ$ under the correspondence $PQR \leftrightarrow YZX$,and $m \angle P = 80^{\circ}$ and $m \angle Q = 40^{\circ}$,find $m \angle Z$. (in $^{\circ}$)

In $\Delta ABC$,$m \angle A = 90^{\circ}$,$AB = 8$ and $AC = 15$. Then,$BC = \ldots$.

In $\Delta ABC$,$m \angle A + m \angle B = m \angle C$. If $AC : BC = 3 : 4$ and $AB = 17.5$,find the perimeter of $\Delta ABC$.