In $\Delta ABC$,$A-D-B$,$A-E-C$ and $\overline{DE} \parallel \overline{BC}$. If $AD = 8$,$AB = 12$ and $AE = 12$,find $EC$.

$O$ is the point of intersection of the diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB \parallel DC$. Through $O$,a line segment $PQ$ is drawn parallel to $AB$ meeting $AD$ in $P$ and $BC$ in $Q$. Prove that $PO = QO$.

$\square ABCD$ is a trapezium in which $\overline{AB} \parallel \overline{CD}$. $M$ is a point on $\overline{AD}$ and $N$ is a point on $\overline{BC}$ such that $AM \times NC = BN \times MD$. If $\overline{MN}$ intersects $\overline{AC}$ at $O$,and $\frac{AM}{MD} = \frac{2}{3}$,find the ratio $\frac{AO}{AC}$.

In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. If $AB = 8$,$AC = 10$ and $BC = 9$,find $BD$ and $DC$.

The lengths of the diagonals of a rhombus are $16 \, cm$ and $12 \, cm$. Then,the length of the side of the rhombus is (in $cm$)

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 $QZ$.