If $\Delta PQR \sim \Delta XYZ$ under the correspondence $PQR \leftrightarrow XYZ$,and given that $3 QR = 4 YZ$ and $PR = 8$,find the length of $XZ$.

In $\Delta ABC$ and $\Delta PQR$,$\angle A \cong \angle R$ and $\angle B \cong \angle P$. If $AB = 6$,$BC = 8$ and $PR = 9$,then $PQ = \ldots$

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude. If $AB = 8$ and $BC = 6$,then $BM = \dots$

The areas of two similar triangles are $144$ and $81$. If the length of the longest side of the first triangle is $36$,the length of the longest side of the second triangle is........

In the given figure,$OB$ is the perpendicular bisector of the line segment $DE$,$FA \perp OB$,and $FE$ intersects $OB$ at the point $C$. Prove that $\frac{1}{OA} + \frac{1}{OB} = \frac{2}{OC}$.

In $\square ABCD$,$\overline{AB} \parallel \overline{CD}$,$M \in \overline{AD}$ and $N \in \overline{BC}$. If $\overline{MN} \parallel \overline{AB}$,prove that $\frac{DM}{MA} = \frac{CN}{NB}$.