Old English wid "vast, broad, long," also used of time, from Proto-Germanic *widaz (source also of Old Saxon, Old Frisian wid, Old Norse viðr, Dutch wijd, Old High German wit, German weit), perhaps from PIE *wi-ito-, from root *wi- "apart, away, in half."
Meaning "distended, expanded, spread apart" is from c. 1500; sense of "embracing many subjects" is from 1530s; meaning "missing the intended target" is from 1580s. As a second element in compounds (such as nationwide, worldwide) meaning "extending through the whole of," is is from late Old English. As an adverb, Old English wide. Wide open "unguarded, exposed to attack" (1915) originally was in boxing, etc. Wide awake (adj.) is first recorded 1818; figurative sense of "alert, knowing" is attested from 1833.
Old English brad "wide, not narrow," also "flat, open, extended," from Proto-Germanic *braidi- (source also of Old Frisian bred, Old Norse breiðr, Dutch breed, German breit, Gothic brouþs), which is of unknown origin. Not found outside Germanic languages. There is no clear distinction in sense from wide. Of day or daylight, late 14c.; of speech or accents, 1530s. Related: Broadly; broadness.