In $\Delta ABC$,$m \angle A = 90^{\circ}$. If $a = 20$ and $b = 12$,then $c = \dots$

Given $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR.$ $\overline{AD}$ is a median in $\Delta ABC$ and $\overline{PM}$ is a median in $\Delta PQR.$ Prove that $\frac{AD}{PM} = \frac{AB}{PQ}.$

In $\Delta ABC$,$m\angle B = 90^{\circ}$,$N \in \overline{AB}$ and $M \in \overline{BC}$. Prove that $AM^{2} + CN^{2} = AC^{2} + MN^{2}$.

$\Delta XYZ \sim \Delta DEF$ for the correspondence $XYZ \leftrightarrow EDF$. If $XY = 3, YZ = 4, ZX = 6$ and $DF = 12$,find the perimeter of $\Delta DEF$.

Below are given the measures of sides $\overline{PQ}$,$\overline{QR}$ and $\overline{PR}$ of $\Delta PQR$. In each case,determine whether $\Delta PQR$ is a right-angled triangle or not. If it is a right-angled triangle,state which angle is a right angle: $PQ = 7, QR = 24, PR = 25$.

In $\Delta ABC$ and $\Delta PQR$,$\angle A \cong \angle P$ and $\angle B \cong \angle Q$. If $\frac{AB}{PQ} = \frac{4}{5}$ and $AC = 20$,find $PR$.