2005 AMC 12B Problems/Problem 14
Problem
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A circle having center , with
, is tangent to the lines
,
and
. What is the radius of this circle?
Solution
Let be the radius of the circle. Draw the two radii that meet the points of tangency to the lines
. We can see that a square is formed by the origin, two tangency points, and the center of the circle. The side lengths of this square are
and the diagonal is
. The diagonal of a square is
times the side length. Therefore,
.