1550s, "coarse fabric," of Scandinavian origin; compare Norwegian dialectal rugga "coarse coverlet," from Old Norse rogg "shaggy tuft," from Proto-Germanic *rawwa-, perhaps related to rag (n.) and rough (adj.). Sense evolved to "coverlet, wrap" (1590s), then "mat for the floor" (1808). Meaning "toupee" is theater slang from 1940. Cut a rug "dance" is slang first attested 1942. To sweep (something) under the rug in the figurative sense is from 1954. Figurative expression pull the rug out from under (someone) "suddenly deprive of important support" is from 1936, American English. Earlier in same sense was cut the grass under (one's) feet (1580s).
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