Basing on the basic concepts and properties of quasihyperbolic mappings and quasisimilarity mappings, to describe the equivalence problem of quasihyperbolic mappings and quasisimilarity mappings in metric space. Using quasihyperbolic metric as main tool to study, the relation between the concepts of maximal stretching and minimal stretching and quasihyperbolic mappings is discussed. It is found that quasihyperbolic mappings and quasisimilarity mappings are equivalent in metric space. The result of research shows that: let be a quasiconvex metric space, Y be a c2-quasiconvex metric space, and let GX and G′Y be two domains. Suppose that f:G→G′ is a homeomorphism, then f is a M-quasihyperbolic mapping if and only if fand f－1are homeomorphism (M1,q)-quasisimilarity mappings, where M1=(c1c2M2(1－αq)－M－1)〖SX(〗α〖〗1－αq〖SX)〗+1, q<min〖JB({〗〖SX(〗1〖〗c1〖SX)〗,〖SX(〗1〖〗c2〖SX)〗〖JB)}〗, α=max{c1,c2}.