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

$\Delta XYZ \sim \Delta DEF$ for the correspondence $XYZ \leftrightarrow EDF$. If $XY = 3, YZ = 4, ZX = 6$ and $DF = 12$,find the perimeter of $\Delta DEF$.

$E$ is a point on the side $AD$ produced of a parallelogram $ABCD$ and $BE$ intersects $CD$ at $F.$ Show that $\Delta ABE \sim \Delta CFB$.

If $\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. If $\text{Area}(\Delta ABC) = 72$,$BC = 6$,and $YZ = 10$,then $\text{Area}(\Delta XYZ) = \dots$

$\Delta PQR \sim \Delta DEF$ for the correspondence $PQR \leftrightarrow DEF$. If $3 PQ = 2 DE$,$EF = 6$,and $PR = 8$,find $QR$ and $DF$.

If $\Delta ABC \sim \Delta XYZ$ under the correspondence $ABC \leftrightarrow XZY$,and the perimeter of $\Delta ABC$ is $45$,the perimeter of $\Delta XYZ$ is $30$,and $AB = 21$,find the length of $XZ$.