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 $\Delta ABC$,$m \angle A = 90^{\circ}$,$AB = 8$ and $AC = 15$. Then,$BC = \ldots$.

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 $\overline{QS}$ is an altitude. If $PS - SR = 10$ and $PQ^2 - QR^2 = 260$,then $PR = \dots$

In $\Delta ABC$,$m \angle B = 90$ and $D$ is the midpoint of $\overline{AC}$. Then,$BD = \dots$

The length of a diagonal of a square is $6$. Then, the length of its sides is $\ldots \ldots \ldots$ (in $\sqrt{2}$)