In rectangle $ABCD$,$AB + BC = 23$ and its area is $120$. If $AB > BC$,then $AB = \ldots$

In $\Delta ABC$,$A-P-B$,$A-Q-C$ and $\overline{PQ} \parallel \overline{BC}$. If $AP = 8x-7$,$PB = 5x-3$,$AQ = 4x-3$ and $QC = 3x-1$,find the value of $x$.

In $\Delta PQR$ and $\Delta XYZ$,$\angle P \cong \angle X$ and $\angle Q \cong \angle Z$. If $PQ = 9$,$QR = 6$,$PR = 4.5$,and $XY = 7.5$,find $YZ$ and $XZ$.

In $\Delta ABC$,$m\angle B = 90^{\circ}$. If $a = 16$ and $c = 12$,then $b = \ldots$

$\Delta XYZ \sim \Delta ABC$ for the correspondence $XYZ \leftrightarrow ABC$. If $\frac{XY}{AB} = \frac{3}{5}$,then $\frac{BC}{YZ} = \dots$

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QM}$ is an altitude to the hypotenuse $\overline{PR}$. If $QM = 4$ and $PR = 10$,find $PM$.