I'm working through identifying triads with roman numerals, but I've come upon one that has stumped me. The triad is the notes E, G, and B and the key is F Major. I believe that it is a 7th(VII) triad because the E is the subtonic tone, and that it is minor because there are three semitones between E and G. That would make it a subtonic minor triad with the roman numeral identifier VII. The answer key I'm working with says that it is a major diminished 7th(vii dim). I'm just starting in music theory so I'm not sure which one is correct, and if I'm wrong, why I'm wrong. To clarify, I'm not trying to find homework answers, I'm just trying to make sure I grasp this concept. Thanks in advance!
You almost had the answer yourself, you said the key is F major. F major has one flat in its key signature, which happens to be the B.
E G B is not a triad you can form using notes only from F major, because F major doesn't have a B natural in it.
So while you're correct that E G B is a minor triad, if you're in F major it's probably E G Bb, which is a diminished triad (root, minor third, diminished fifth).
Don't forget when you're looking at this kind of construction that those default triads have to be made out of the notes of the diatonic scale which will include any sharps and flats in the key signature. All too easy to do!