mathschallenge.net

Problem

A unit circle is placed against a right angle.

What is the radius of the smaller circle?

Solution

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?