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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 minus r)2 + (1 minus r)2 = 2(1 minus r)2
therefore 1 + 2r + r2 = 2(1 minus 2r + r2) = 2 minus 4r + 2r2
therefore r2 minus 6r + 1 = 0

Solving the quadratic we get r = 3 plus or minus 2radical2, giving the solution r = 3 minus 2radical2.

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

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