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