Physical sizes of extended radio galaxies can be employed as a cosmological "standard ruler," using a previously developed method. Eleven new radio galaxies are added to our previous sample of 19 sources, forming a sample of 30 objects with redshifts between 0 and 1.8. This sample of radio galaxies are used to obtain the best-fit cosmological parameters in a quintessence model in a spatially flat universe, a cosmological constant model that allows for nonzero space curvature, and a rolling scalar field model in a spatially flat universe. Results obtained with radio galaxies are compared with those obtained with different supernova samples, and with combined radio galaxy and supernova samples. Results obtained with different samples are consistent, suggesting that neither method is seriously affected by systematic errors. Best-fit radio galaxy and supernovae model parameters determined in the different cosmological models are nearly identical, and are used to determine dimensionless coordinate distances to supernovae and radio galaxies, and distance moduli to the radio galaxies. The distance moduli to the radio galaxies can be combined with supernovae samples to increase the number of sources, particularly high-redshift sources, in the samples. The constraints obtained here with the combined radio galaxy plus supernovae dataset in the rolling scalar field model are quite strong. The best-fit parameter values suggest that Ω _m is less than about 0.35, and the model parameter α is close to zero; that is, a cosmological constant provides a good description of the data. We also obtain new constraints on the physics of engines that power the large-scale radio emission. The equation that describes the predicted size of each radio source is controlled by one model parameter, β, which parameterizes the extraction of energy from the black hole. Joint fits of radio galaxy and supernova samples indicate a best-fit value of β that is very close to a special value for which the relationship between the braking magnetic field strength and the properties of the spinning black hole is greatly simplified, and the braking magnetic field strength depends only upon the spin angular momentum per unit mass and the gravitational radius of the black hole. The best-fit value of β of 1.5 indicates that the beam power L_j and the initial spin energy of the black hole E are related by L_j E ², and that the relationship that might naively be expected for an Eddington limited system, L_j E, is quite clearly ruled out for the jets in these systems.