Draw the two chords. Let O be the center of the circle.  Draw OC 
perpendicular to both chords. That divides the two chords in half.
So CD = 10 and AB = 14.  Draw radii OA and OD, both equal to radius r.
We are given that BC = 5, the distance between the two chords.  Let
OB = x.

We use the Pythagorean theorem on right triangle ABO

AO² = AB² + OB²
 r² = 14² + x²

We use the Pythagorean theorem on right triangle DCO

DO² = CD² + OC²

We see that OC = OB+BC = x+5, so

 r² = 10² + (x+5)²

So we have a system of two equations:

  r² = 14² + x²
  r² = 10² + (x+5)²

Since both left sides equal r², set the right sides
equal to each other. 

 14² + x² = 10² + (x+5)²
 196 + x² = 100 + x² + 10x + 25
      196 = 125 + 10x
       71 = 10x
      7.1 = x

       r² = 14² + x²
       r² = 196 + (7.1)²
       r² = 196 + 50.41
       r² = 246.41
        r = √246.41
        r = 15.69745202 cm 

Edwin