Rent
Middle English: from Old French rente, from a root shared by render.
wiktionary
From Middle English rent, rente, from Old English renta, from Old French rente and Medieval Latin renta, both from Vulgar Latin *rendere, from Latin reddere, present active infinitive of reddō.
From Middle English renten(“to tear”). Variant form of renden.
etymonline
rent (n.1)
[payment for use of property], mid-12c., in a legal sense, "compensation made periodically, with reference to time of possession and use of property," from Old French rente "payment due; profit, income" and Medieval Latin renta, both from Vulgar Latin *rendita, noun use of fem. past participle of *rendere "to render" (see render (v.)).
It came into English earlier in a more general, and now obsolete, sense of "income, revenue" (late Old English). The sense in political economy, "what is left from the produce of the soil after deducting what is necessary to support the producers, interest, seed-corn, etc.," is by 1815. Rent-free is attested from 1630s.
Rents (to think how much of evil there is in the two senses of that four-lettered word ! In the two methods of intonation of its synonym, Tear !) [Ruskin, "Fors Claveriga"]
rent (n.2)
"torn place, opening made by rending or tearing," 1530s, noun use of Middle English renten "to tear, rend" (early 14c.), a variant of renden (see rend (v.)). Of clefts or fissures in the earth by 1702.
rent (v.)
mid-15c., renten, "to rent out property, grant possession and enjoyment of in exchange for a consideration paid in the nature of rent," from Old French renter "pay dues to," or from rent (n.1). Related: Rented; renting.
Earlier (mid-14c.) it was used in the more general sense of "provide with revenue, endow with income." The sense of "to take and hold in exchange for rent" is from 1520s. The intransitive sense of "be leased for rent" is from 1784.
Prefix rent-a- is attested by 1921, mainly of businesses that rented various makes of car (Rentacar is a trademark registered in U.S. 1924); extended to other "temporary" uses since 1961.
