解:?$(1)$?∵?$C$?為線段?$AB$?的中點(diǎn),且?$AB=10\ \mathrm {cm}$?
∴?$AC=BC=\frac {1}{2}AB=\frac {1}{2}×10=5(\mathrm {cm})$?
∵?$M$?為線段?$AC$?的中點(diǎn),?$N$?為線段?$BC$?的中點(diǎn)
∴?$MC=\frac {1}{2}AC=\frac {5}{2}(\mathrm {cm})$?,?$NC=\frac {1}{2}BC=\frac {5}{2}(\mathrm {cm})$?
∴?$MN=MC+NC=5(\mathrm {cm})$?
?$(2)$?∵?$AC∶BC=3∶2$?,且?$AB=a$?
∴?$AC=\frac {3}{5}AB=\frac {3}{5}a$?,?$BC=\frac {2}{5}AB=\frac {2}{5}a$?
∵?$M$?為線段?$AC$?的中點(diǎn),?$N$?為線段?$BC$?的中點(diǎn)
∴?$MC=\frac {1}{2}AC=\frac {3}{10}a$?,?$NC=\frac {1}{2}BC=\frac {1}{5}a$?
∴?$MN=MC+NC=\frac {3}{10}a+\frac {1}{5}a=\frac {1}{2}a$?