Don't think Left Hand, Right Hand think "Chord"
C-F#/????
You could play:
C-F# / Ab Eb - to make it an Ab7 Chords
C-F# /A-E - to make it an Am6 Chord
C-F# / A E - To make it an Ab#9#5 Chords
There are tons of options. The key is to understand the basics, so you can understand where it can be used. the word "Tri-tone" is a very vague term.