In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 12$ and $CM = 3$,find $BM$.

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 $AC$.

In $\Delta PQR$,$m \angle P = 90^{\circ}$ and $\overline{PM}$ is an altitude. If $QM = 2x^{2}$ and $RM = 8x^{2}$,then $PM = \ldots \ldots$

Two sides and the perimeter of one triangle are respectively three times the corresponding sides and the perimeter of the other triangle. Are the two triangles similar? Why?

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 $\Delta ABC$,$\overline{AB} \cong \overline{AC}$ and $\overline{AD}$ is a median. If $AD = 12$ and the perimeter of $\Delta ABC$ is $36$,find the area of $\Delta ABC$.