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Old English rap "rope, cord, cable," from Proto-Germanic *raipaz (cf. Old Norse reip, West Frisian reap, Middle Dutch, Dutch reep "rope," Old Frisian silrap "shoe-thong," Gothic skauda-raip "shoe-lace," Old High German, German reif "ring, hoop"). Technically, only cordage above one inch in circumference and below 10 (bigger-around than that is a cable). Nautical use varies. Finnish raippa "hoop, rope, twig" is a Germanic loan-word.
To know the ropes (1840, Dana) originally is a seaman's term. Phrase on the ropes "defeated" is attested from 1924, a figurative extension from the fight ring, where ropes figure from 1829. To be at the end of (one's) rope "out of resources and options" is first attested 1680s. Formerly also in many slang and extended uses related to punishment by hanging, e.g. John Roper's window "a noose," rope-ripe "deserving to be hanged," both 16c. To give someone (enough) rope (to hang himself) is from 1650s.
c.1300, "bind with a rope," from rope (n.). Meaning "mark off with rope" is from 1738; to rope (someone or something) in is from 1848. Related: Roped; roping.