So it seems to me that the main issue here is that, considering the circular arc, you don't know what angle the arc describes. But you do know the length of the chord corresponding to the arc, and you know the arc length. Let me outline one method of solution.
1. You know that the chord length is given by
2. Since you now have two equations in two unknowns, solving for and
3. Call the desired distance and the perpendicular from the center of the circle to the chord has length Then
4. From basic trigonometry, you also know that From this you can get
5. Since you are now done.