jmc

By SAS Similarity, $△ABC\sim △MNC$, so $AB\parallel MN$. Thus, by AA Similarity, $△AGB\sim △NGM$. Let $a=GN$ and $b=GM$, so $AG=2a$ and $BG=2b$. By the Pythagorean Theorem,$$4a^2+b^2=9$$$$a^2+4b^2=\frac{49}{4}$$Adding the two equations yields $5a^2+5b^2=\frac{85}{4}$, so $a^2+b^2=\frac{17}{4}$. Thus, $MN=\frac{\sqrt{17}}{2}$, so $AB=\boxed{\sqrt{17}}$.