We study asymptotics of perfect matchings on a large class of graphs called the contracting square-hexagon lattice,whichis constructed row by row from either a row of a square grid or a row of a hexagonal lattice. We assign thegraph periodicedge weights with period 1*n, and consider the probability measure of perfect matchings in which theprobability of eachconfiguration is proportional to the product of edge weights. We show that the partition function ofperfect matchings onsuch a graph can be computed explicitly by a Schur function depending on the edge weights. Byanalyzing the asymptoticsof the Schur function, we then prove the Law of Large Numbers (limit shape) and the CentralLimit Theorem (convergenceto the Gaussian free field) for the corresponding height functions. We also show that thedistribution of certain type ofdimers near the turning corner is the same as the eigenvalues of Gaussian Unitary Ensemble,and explicitly study thecurve separating the liquid region and the frozen region for certain boundary conditions.