Let the radius of the table be r.

Denote the corner of the room (0,0), the point of contacts (0,r) and (r,0) and the point on the edge in question (10,5).

Draw a triangle with these points (10,5), (r,5), and (r,r). This is a right triangle with sides of length r-10, r-5, and r. The pythagorean formula tells us that (r-10)2 + (r-5)2 = r2. Solving for r yields r=5 and 25.

Thanks to Nick Hobson for this one.

Michael Shackleford, A.S.A.