I always find it easier to think of half diminished chords as minor7flat5, so for B, think of a Bm7 and flat the 5.
B F A D ... that would be the vii chord of the key of C. So it's the F# of Bm (the 5) that gets flatted down to F.
(A half-diminished chord is 1-b3-5b-b7 ... a minor 7th chord is 1-b3-5-b7, so the only difference between the two is that 5. Look at any minor7th chord shape, find the 5(s) within and flat it(them) ... now you have a half-diminished chord. Simple!)
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