In $\Delta ABC$,$m \angle B = 90^\circ$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. If $BC = 2x + 2$,$CM = x + 1$,and $AC = 5x$,find the value of $x$.

In $\Delta ABC$ and $\Delta PQR$,if $\frac{AB}{PQ} = \frac{BC}{PR} = \frac{CA}{QR}$,then the correspondence $ABC \leftrightarrow \dots$ is a similarity.

In $\Delta ABC$,$m \angle A = 90^{\circ}$ and $\overline{AM}$ is an altitude to the hypotenuse $\overline{BC}$. If $BM = 6$ and $CM = 2$,find the perimeter of $\Delta ABC$.

In $\Delta PQR$, $m \angle Q = 90^{\circ}$ and $\overline{QM}$ is an altitude to the hypotenuse $PR$. If $QM = 14$ and $RM = 7$, find $PQ$. (in $\sqrt{5}$)

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 $XZ$.

$A$ ladder is placed leaning on a wall. Its upper end reaches to the height of $12 \, m$ on the wall and its lower end rests $9 \, m$ away from the base of the wall. Find the length of the ladder in meters.