showroom (n.)

Middle English sheuen, from Old English sceawian "to look at, see, gaze, behold, observe; inspect, examine; look for, choose," from Proto-Germanic *skauwojanan (source also of Old Saxon skauwon "to look at," Old Frisian skawia, Dutch schouwen, Old High German scouwon "to look at"), from Proto-Germanic root *skau- "behold, look at," from PIE *skou-, variant of root *keu- "to see, observe, perceive."

The causal meaning "let be seen; put in sight, make known" evolved c. 1200 for unknown reasons, seems to be unique to English (German schauen still means "look at"), and in a century displaced the older meaning. The sense of "explain, make clear" is from c. 1300, as the intransitive sense of "be seen, appear."

The spelling shew, popular 18c. and surviving into early 19c., represents an obsolete pronunciation (rhymes with view). The horse-racing sense of "finish third or in the top three" is by 1903, perhaps from an earlier sense in card-playing.

Middle English roum, from Old English rum "space, extent; sufficient space, fit occasion (to do something)," from Proto-Germanic *ruman (source also of Old Norse, Old Saxon, Old High German, Gothic rum, German Raum "space," Dutch ruim "hold of a ship, nave"), nouns formed from Germanic adjective *ruma- "roomy, spacious," from PIE root *reue- (1) "to open; space" (source also of Avestan ravah- "space," Latin rus "open country," Old Irish roi, roe "plain field," Old Church Slavonic ravinu "level," Russian ravnina "a plain").

Old English also had a frequent adjective rum "roomy, wide, long, spacious," also an adverb, rumlice "bigly, corpulently" (Middle English roumli).

The meaning "chamber, cabin" is recorded by early 14c. as a nautical term; applied by mid-15c. to interior division of a building separated by walls or partitions; the Old English word for this was cofa, ancestor of cove. The sense of "persons assembled in a room" is by 1712.

Make room "open a passage, make way" is from mid-15c.  Room-service is attested from 1913; room-temperature, comfortable for the occupants of a room, is so called from 1879. Roomth "sufficient space" (1530s, with -th (2)) now is obsolete.