In $\Delta ABC$,$m \angle B = 90^\circ$. If $AC - AB = 27$ and $AC - BC = 6$,find the perimeter of $\Delta ABC$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$,$\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$,and $AM > CM$. If $BM = 12$ and $AC = 25$,find $AB$ and $BC$.

Point $O$ lies in the interior of $\Delta PQR$. Given $O-A-P$,$O-B-Q$,$O-C-R$,$\overline{AB} \parallel \overline{PQ}$ and $\overline{BC} \parallel \overline{QR}$. Prove that $\overline{AC} \parallel \overline{PR}$.

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QM}$ is an altitude. If $PQ = 40$ and $QR = 30$,find the area of $\Delta QMR$.

If $\Delta ABC \sim \Delta DEF$ for the correspondence $ABC \leftrightarrow DEF$,and given $AB = 8$,$AC = 10$,$DE = 12$,and $EF = 18$,find the perimeter of $\Delta DEF$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM - MC = 7$ and $AB^{2} - BC^{2} = 175$,then $AC = \ldots$