Given $\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. If $AB : XY = 4 : 5$ and $YZ = 20$,find $BC$.

In $\Delta XYZ$,$m\angle Y = 90^{\circ}$. If $XY = a^{2} - b^{2}$ and $YZ = 2ab$,find $XZ$ (where $a > b > 0$).

In $\Delta ABC$,$m \angle B = 90^{\circ}$. If $AB = 6$ and $AC = 10$,find the area of $\Delta ABC$.

In $\Delta ABC$,$AB = 3$,$BC = 6$,and $AC = 8$. In $\Delta PQR$,$PQ = 13.6$,$QR = 10.2$,and $PR = 5.1$. Which correspondence between $\Delta ABC$ and $\Delta PQR$ is a similarity?

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $m \angle A : m \angle C = 1 : 2$. If $BC = 4$,find $AC$.

Transversals $t_{1}$ and $t_{2}$ to $l_{1} \parallel l_{2} \parallel l_{3}$ intersect them at $A, B, C$ and $P, Q, R$ respectively. If $AB = 5$,$BC = 8$ and $PQ = 6.5$,find $QR$.