The Carnegie Chicago Hubble Program X: Tip of the Red Giant Branch Distances to NGC 5643 and NGC 1404

The primary goal of the Carnegie Chicago Hubble Program (CCHP) is to calibrate the zero-point of the Type Ia supernova (SN Ia) Hubble Diagram through the use of Population II standard candles. So far, the CCHP has measured direct distances to 11 SNe Ia, and here we increase that number to 15 with two new TRGB distances measured to NGC 5643 and NGC 1404, for a total of 20 SN Ia calibrators. We present resolved, point-source photometry from new Hubble Space Telescope (HST) imaging of these two galaxies in the F814W and F606W bandpasses. From each galaxy's stellar halo, we construct an F814W-band luminosity function in which we detect an unambiguous edge feature identified as the Tip of the Red Giant Branch (TRGB). For NGC 5643, we find $\mu_0 = 30.48\pm0.03(stat)\pm0.07(sys) $ mag, and for NGC 1404 we find $ \mu_0=31.36\pm 0.04(stat)\pm 0.05(sys)$ mag. From a preliminary consideration of the SNe Ia in these galaxies, we find increased confidence in the results presented in Paper VIII (Freedman et al. 2019). The high precision of our TRGB distances enables a significant measurement of the 3D displacement between the Fornax Cluster galaxies NGC 1404 and NGC 1316 (Fornax A) equal to $1.50^{+0.25}_{-0.39}$ Mpc, which we show is in agreement with independent literature constraints.