In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = x+1$,$MB = 2x-2$,$AN = 2x+2$ and $NC = 3x+1$,find the value of $x$.

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

In $\Delta ABC$,medians $\overline{AD}$ and $\overline{BE}$ intersect at $G$. $A$ line through $G$ and parallel to $\overline{DE}$ intersects $\overline{AC}$ at $K$. If $EK = 1.8$,then $AC = \ldots$

In $\Delta PQR$,the bisector of $\angle P$ intersects $\overline{QR}$ at $S$. If $QS = 2$,$PQ = 5$ and $SR = 3$,find $PR$.

In $\Delta ABC,$ the midpoints of the sides are $P, Q, R.$ In $\Delta PQR,$ the midpoints of the sides are $X, Y, Z.$ If the area of $\Delta XYZ$ is $20,$ find the area of $\Delta PQR$ and $\Delta ABC.$

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude. If $BM = 12$ and $AM = 9$,then $AC = \ldots$