In $\Delta XYZ$,$\overline{XM}$ is a median. If $XY = 11$ and $XM = YM = 30.5$,find $XZ$.

In $\Delta ABC$,$A-P-B$,$A-Q-C$ and $\overline{PQ} \parallel \overline{BC}$. If $AP = 2$,$AB = 6$ and $AC = 9$,find $AQ$.

In $\Delta PQR$ and $\Delta XYZ$,if $\frac{PQ}{XY} = \frac{QR}{YZ} = \frac{PR}{XZ} = \frac{2}{5}$,then the ratio of their areas $\text{Area}(\Delta PQR) : \text{Area}(\Delta XYZ) = \ldots$

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 $\Delta ABC$,$m \angle B = 90^{\circ}$ and $m \angle A : m \angle C = 1 : 2$. If $BC = 4$,find $AC$.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM : AB = 2 : 3$ and $AC = 15$,then $NC = \ldots$