This is a brilliant question! After doing a little research, I found the answer.
Firstly, the word 'audio' comes from the Latin verb 'audire', which means 'to hear'. This was subsequently shortened to 'audio', which can be used as a prefix (e.g. 'audiometry', which is the measurement of the range and sensitivity of a person's hearing). More recently, 'audio' can also be used as a word on its own, describing sound devices.
The word 'video', similarly comes from Latin - from the verb 'videre', which means 'to see'.
(This makes sense considering audio media concerns hearing and listening, and video media concerns seeing and watching.)
Unfortunately - the answer to your interesting question is quite boring. The word 'audio' came about before the word 'video'. When the word 'video' was devised in the 1930s, it was constructed on the template of 'audio'.
Both of these original Latin verbs end with '-re', however 'audire' has an 'i' before the end, and 'videre' has an 'e' before the end.
So, as 'audire' took the beginning of the word:
'aud'
And the third to last letter of the word:
'i'
And the last letter was 'o', this gave:
aud + i + o = audio
Thus, 'videre' took the beginning of the word:
'vid'
And the third to last letter of the word:
'e'
And the last letter 'o', to give:
vid + e + o = video
Great question.
Hope this helps!
P.S. The word 'radio' actually has a different origin entirely (which I didn't know until just then!) It was shortened from words like 'radiotelegraph' and 'radiotelephone' in about 1915. These words came from the Latin word 'radius', which means 'beam or ray' - referring to the use of radio waves for the transmission of sound (these are a form of electromagnetic/light wave, hence 'beam' and 'ray').
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