Summary: We point out the relation between the orthogonal polynomials with respect to (w.r.t.) varying measures and the so-called orthogonal rationals on the unit circle in the complex plane. This observation enables us to combine different techniques in the study of these polynomials and rationals. As an example, we present a simple and short proof for a known result on the weak-star convergence of orthogonal polynomials w.r.t. varying measures. Some related problems are also considered.