A unit circle is placed against a right angle.
What is the radius of the smaller circle?
Let the radius of the small circle be r.
Using the Pythagorean Theorem, (1 + r)2 = (1 r)2 + (1 r)2 = 2(1 r)2
1 + 2r + r2 = 2(1 2r + r2) = 2 4r + 2r2
r2 6r + 1 = 0
Solving the quadratic we get r = 3 22, giving the solution r = 3 22.
If a unit sphere is placed in the corner of a room, what is the largest sphere that can be placed in the gap between the unit sphere and the walls?