Solution:

Place the center at height $0.$ Two equal chords lie at equal distances from the center, so the three equally spaced lines are at heights $-\tfrac{d}{2},\tfrac{d}{2},\tfrac{3d}{2},$ with the two $38$-chords at $\pm\tfrac{d}{2}$ and the $34$-chord at $\tfrac{3d}{2}.$

Half-chord relations give $r^2-\left(\tfrac{d}{2}\right)^2=19^2$ and $r^2-\left(\tfrac{3d}{2}\right)^2=17^2.$

Subtracting, $2d^2=19^2-17^2=72,$ so $d^2=36$ and $d=6.$

Thus, the correct answer is B.