A full quantum mechanical approach is extended to a coupled waveguide-ring resonator interacting with a three-level quantum dot (QD). The transmission and reflection amplitudes, as well as the resonator and the QD responses, are solved exactly via a real-space approach, which shows the single-photon transport properties. One feature of the system is the coupling between the two degenerate whispering- gallery modes (WGMs) of the resonator at rate β due to backscattering caused by surface roughness of the cavity. Another feature is describing the coupling between QD and the cavity by two parameters g1 and g2. From the analytic numerical results we can see the difference from two-level system.It may better interpret the experiment of a quantum dot strongly coupled to WGMs in a cavity.