In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = 3x - 1$,$MB = 2x + 1$,$AN = 3x + 1$ and $NC = 5x + 1$,find the value of $x$.

In $\Delta ABC$,$D$,$E$,and $F$ are the midpoints of $\overline{AB}$,$\overline{BC}$,and $\overline{CA}$ respectively. Then the correspondence $DEF \leftrightarrow \dots$ is a similarity.

$\Delta ABC$ and $\Delta XYZ$ are similar under the correspondence $ABC \leftrightarrow XZY$. If $AB = 6$,$XY = 12$,$YZ = 6$,and $ZX = 9$,find the perimeter of $\Delta ABC$.

In $\Delta ABC$,$D$ is the midpoint of $\overline{BC}$. $\Delta ABC$ and $\Delta BDE$ are equilateral triangles. Then,the ratio of the areas of $\Delta ABC$ and $\Delta BDE$ is.......

In $\Delta ABC$,$\overline{AD}$ and $\overline{BE}$ are medians. Line $m$ through $D$ and parallel to $\overline{BE}$ intersects $\overline{AC}$ at $K$. If $EK = 2.5$,then $AC = \ldots$

In $\Delta XYZ$,$X-P-Y$,$X-Q-Z$ and $\overline{PQ} \parallel \overline{YZ}$. If $XP:PY = 3:5$ and $XZ = 5.6$,find $QZ$.