$\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. If $AB = 8$,$PQ = 14$ and the perimeter of $\Delta ABC$ is $20$,find the perimeter of $\Delta PQR$.

In $\Delta ABC$,$m \angle A + m \angle C = m \angle B$ and $AC : AB = 17 : 15$. If $BC = 8$,then the area of $\Delta ABC$ is.............

$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. $\overline{AD} \perp \overline{BC}$,$D \in \overline{BC}$ and $\overline{XM} \perp \overline{YZ}$,$M \in \overline{YZ}$. Prove that $\frac{AD}{XM} = \frac{BC}{YZ}$.

In $\Delta ABC$,$m\angle B = 90^{\circ}$,$AB = 2x + 3$,$BC = x + 2$,and $AC = 3x - 1$. Find the value of $x$.

Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides,then the two sides are divided in the same ratio.

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........