\[\begin{array}{l} {\text{а)}} \hfill \\ CD = DO = DM \Rightarrow \angle DMC = \angle MCD = \frac{\pi }{4} \hfill \\ \vartriangle CND = \vartriangle CMD \Rightarrow CM = CN \hfill \\ \angle NCM = \angle NCD + \angle MCD = \frac{\pi }{2} \hfill \\ \vartriangle CMN \sim \vartriangle ABC \Rightarrow AC = BC \hfill \\ {\text{б)}} \hfill \\ {\text{Пусть }}DQ = x \hfill \\ \vartriangle DQM \sim \vartriangle AQB,{\text{ }}k = \frac{{AB}}{{DM}} = 6 \Rightarrow AQ = 6x \hfill \\ AD = 5x \hfill \\ A{D^2} = A{H^2} + D{H^2} \Leftrightarrow x = \frac{{\sqrt {13} }}{2}. \hfill \\ \end{array}\]

б) \[3\sqrt {13} \]