4. In $\triangle A B C$, if $|\overrightarrow{A B}|=2,|\overrightarrow{A C}|=3,|\overrightarrow{B C}|=4, O$ is the incenter of $\triangle A B C$, and $\overrightarrow{A O}=\lambda \overrightarrow{A B}+\mu \overrightarrow{B C}$, then $\lambda+\mu=$
A. $\frac{9}{10}$
B. $\frac{7}{10}$
C. $\frac{8}{9}$
D. $\frac{7}{9}$
Extend $A O$ to intersect $B C$ at $D$, then $B D=\frac{8}{5}, D C=\frac{12}{5} \Rightarrow \overrightarrow{A O}=\frac{5}{9} \overrightarrow{A D}$, and $\overrightarrow{B D}=\frac{2}{5} \overrightarrow{B C} \Rightarrow \overrightarrow{A D}=\frac{3}{5} \overrightarrow{A B}+\frac{2}{5} \overrightarrow{A C}=\overrightarrow{A B}+\frac{2}{5} \overrightarrow{B C}$, thus $\overrightarrow{A O}=\frac{5}{9} \overrightarrow{A B}+\frac{2}{9} \overrightarrow{B C} \Rightarrow \lambda+\mu=\frac{7}{9}$, so the answer is $D$.