## Corner Circle

#### 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 + 2`r` + `r`^{2} = 2(1 2`r` + `r`^{2}) = 2 4`r` + 2`r`^{2}

`r`^{2} 6`r` + 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?

Problem ID: 84 (Oct 2002) Difficulty: 3 Star