$\begin{array}{l} \pi < \alpha < \dfrac{{3\pi }}{2} \to \sin \alpha < 0,\cos \alpha < 0\\ \left\{ \begin{array}{l} {\sin ^2}\alpha + {\cos ^2}\alpha = 1\\ \sin 2\alpha = 2\sin \alpha \cos \alpha = \dfrac{7}{9} \end{array} \right. \Rightarrow {\left( {\sin \alpha + \cos \alpha } \right)^2} = \dfrac{{16}}{9} \Rightarrow \sin \alpha + \cos \alpha = \dfrac{{ - 4}}{3}\\ M = \sqrt {{{\cos }^2}\alpha - 4\cos \alpha + 4} + \sqrt {{{\sin }^2}\alpha - 4\sin \alpha + 4} \\ M = \left| {\cos \alpha - 2} \right| + \left| {\sin \alpha - 2} \right| = 4 - \left( {\sin \alpha + \cos \alpha } \right) = 4 + \dfrac{4}{3} = \dfrac{{16}}{3}\\ \end{array}$
