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 + 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?