5. In Rt $\triangle A B C$, $C D$ is the altitude on the hypotenuse $A B$, $A C=3, B C=4$, and $r, r_{1}, r_{2}$ represent the radii of the inscribed circles of $\triangle A B C, \triangle A C D, \triangle B C D$ respectively. Then $r+r_{1}+r_{2}=(\quad)$.
(A) $\frac{12}{5}$
(B) 3
(C) 4
(D) 5
5. A.

As shown in Figure 4, by the Pythagorean theorem,
$$
A B=5 \text {, }
$$

By the area relationship,
$$
C D=\frac{12}{5} .
$$

The inradius of the right triangle $\triangle A B C$ is $r=\frac{1}{2}(C A+C B-A B)$.
Similarly, $r_{1}=\frac{1}{2}(D C+D A-C A)$,
$$
r_{2}=\frac{1}{2}(D C+D B-C B) \text {. }
$$

Thus, $r+r_{1}+r_{2}=D C=\frac{12}{5}$.