In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude. If $BM = 12$ and $AM = 9$,then $AC = \ldots$

$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. $\overline{AD} \perp \overline{BC}$,$D \in \overline{BC}$ and $\overline{XM} \perp \overline{YZ}$,$M \in \overline{YZ}$. Prove that $\frac{AD}{XM} = \frac{BC}{YZ}$.

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 $\Delta DEF$.

Prove that the sum of the squares of the sides of a rectangle is equal to the sum of the squares of its diagonals.

In $\Delta ABC$,$m\angle B = 90^\circ$ and $\overline{BM}$ is an altitude. If $AB = 8$ and $BC = 6$,find $AM$,$BM$,and $CM$.

$\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow YZX$. If $m \angle P = 2 m \angle Q$ and $m \angle X = 120^\circ$,then $m \angle Y = \dots$ (in $^\circ$)