$\Delta DEF \sim \Delta PQR$ for the correspondence $DEF \leftrightarrow QPR$. If $2 DE = 3 PQ$ and $QR = 8$,then $DF = \ldots$

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude. If $AB = \sqrt{10}$ and $AM = 2.5$,then $MC = \ldots$

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

If $\Delta ABC \sim \Delta DEF$ for the correspondence $ABC \leftrightarrow DEF$,and given $AB = 8$,$AC = 10$,$DE = 12$,and $EF = 18$,find the perimeter of $\Delta DEF$.

In $\Delta ABC$,$m \angle A = m \angle B + m \angle C$ and $\overline{AM}$ is an altitude. If $AM = \sqrt{12}$ and $BC = 8$,find $BM$.

In $\Delta ABC$,$\overline{AD}$ is a median and $AB^2 + AC^2 = 148$. If $AD = 7$,then $BC = \ldots$