If $\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. If $\text{Area}(\Delta ABC) = 72$,$BC = 6$,and $YZ = 10$,then $\text{Area}(\Delta XYZ) = \dots$

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = 6$,$MB = 12$ and $AN = 8$,find $AC$.

In $\Delta ABC$,$D \in \overline{AB}$,$E \in \overline{AC}$ and $\overline{DE} \parallel \overline{BC}$. $F$ is a point on $\overline{AD}$ such that $\overline{EF} \parallel \overline{CD}$. Prove that $AD^2 = AB \times AF$.

Find the value of $x$ for which $DE \parallel AB$ in the given figure.

$\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow RPQ$. If $m \angle A + m \angle C = m \angle B$,then in $\Delta PQR$,$\ldots \ldots \ldots$ is a right angle.

$O$ 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 $PO = QO$.