(Reuters) – New U.S. home-building activity fell to a nine-month low in June and permits for new construction projects slipped as well, the latest indication of a cooling housing market as surging mortgage rates reduce affordability. While multi-family construction gained ground as rising rents burnish the appeal of apartment projects, cushioning the overall decline, activity in the single-family segment dropped to a two-year low. Housing overall is set to have been a drag on U.S. gross domestic product in the second quarter. Housing starts fell 2% to a seasonally adjusted annual rate of 1.559…