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.