Call the radius "r".
Draw the 48 cm chord. Draw radii to both endpoints of the chord, and a third radius that is the perpendicular bisector to the chord. This forms two congruent triangles. They have sides r (the hypotenuse), x (the distance from the center of the circle to the chord, and 24.
Now do the same thing for the other chord. It will have sides r (the hypotenuse), x+8 (the distance from the center of the circle to the chord, and 20.
Now we use the Pythagorean theorem twice:
r^2 = (x+8)^2 + 20^2
r^2 = x^2 + 24^2
Since thee two equations are both equal to r^2, we can just write one equation with "x" in it:
(x+8)^2 + 400 = x^2 + 576
now expand the left hand side (x+8)^2 term using FOIL or whatever you like:
(x^2 + 16x + 64) + 400 = x^2 + 576
subtract x^2 and 464 from both sides:
16x = 112
Divide each side by 7:
x = 7
Your two triangles are a 7-24-r triangle and a 15-20-r triangle, where r is the hypotenuse. Pick your favorite one and you'll see that r, the radius of the circle that we needed to find, is 25 cm. (15-20-25 is just a 3-4-5 triangle stretched by a factor of 5. 7-24-25 is a pythagorean triple all by itself).