[tex]\displaystyle
\text{A-----------------------M-----------------------B--------N---------C} \\ \\
AB=9\;cm \\ \\
AB = \frac{3}{4}AC \\ \\
AC = AB : \frac{3}{4} = AB \times \frac{4}{3} = 9 \times \frac{4}{3} = 12\;cm \\ \\
BC = 12 - 9 = 3\; cm \\ \\
MB = \frac{AB}{2} = \frac{9}{2} \;cm \\ \\
BN = \frac{BC}{2} = \frac{3}{2} \;cm \\ \\
MN = MB+BN = \frac{9}{2}+\frac{3}{2}=\frac{9+3}{2} =\frac{12}{2}=\boxed{6\;cm}[/tex]
