videotape (n.)

1935, as visual equivalent of audio, from Latin video "I see," first person singular present indicative of videre "to see" (see vision). As a noun, "that which is displayed on a (television) screen," 1937.

Engineers, however, remember the sad fate of television's first debut and are not willing to allow "video transmission" (as television is now called by moderns) to leave the laboratory until they are sure it will be accepted. [The Michigan Technic, November 1937]

video game is from 1973.

Old English tæppe "narrow strip of cloth used for tying, measuring, etc.," of uncertain origin; perhaps [Klein] a back-formation from Latin tapete "cloth, carpet," compare also Old Frisian tapia, Middle Low German tapen "to pull, pluck, tear." The original short vowel became long in Middle English.

Adhesive tape is from 1885; also in early use sometimes friction tape. Tape recorder "device for recording sound on magnetic tape" first attested 1932; from earlier meaning "device for recording data on ticker tape" (1892), from tape in the sense of "paper strip of a printer" (1884). Tape-record (v.) is from 1950. Tape-measure is attested from 1873; tape-delay is from 1968.