$P\hat{M}Q={{90}^{{}^\circ }}\Rightarrow M{{P}^{2}}+M{{Q}^{2}}=P{{Q}^{2}}$ $\left\{ \begin{matrix} MQ\Rightarrow \frac{AQ}{QB}=\frac{AM}{MB}  \\ MP\Rightarrow \frac{AP}{PC}=\frac{AM}{MC}  \\ \end{matrix} \right.\xrightarrow{MB=MC}\frac{AQ}{QB}=\frac{AP}{PC}\Rightarrow PQ\left\| BC \right.\Rightarrow \frac{AP}{AC}=\frac{PQ}{BC}(1)$ $\frac{AP}{PC}=\frac{AM}{MC}\Rightarrow \frac{AP}{PC}=\frac{3}{5}\Rightarrow \frac{AP}{AP+PC}=\frac{3}{3+5}\Rightarrow \frac{AP}{AC}=\frac{3}{8}\xrightarrow{(1)}\frac{PQ}{BC}=\frac{3}{8}$ $\Rightarrow \frac{PQ}{10}=\frac{3}{8}\Rightarrow PQ=\frac{15}{4}\Rightarrow M{{P}^{2}}+M{{Q}^{2}}=P{{Q}^{2}}={{(\frac{15}{4})}^{2}}=\frac{225}{16}$