| Inscribed Circle |
A circle C is inscribed in a square with sides of length 4 inches.
A second circle O is tangent to the square in exactly two places, and is also tangent to circle C.
What is the radius of circle O? |
Problem Moderated by: Douglas |
| Problem Solution |
In the diagram, the vertical blue line is equal to the radius of the circle O. Since the triangle formed by the red line, the vertical blue line, and the base of the square is a 45-45-90 triangle, this means the length of the red line is r(sqrt(2)).
Since the radius of circle C is 2, the length of the diagonal line, from the corner to of the square to the center of C, is
r(sqrt(2)) + r + 2
But this is half the diagonal of the square, or 2sqrt(2).
Thus r(sqrt(2)) + r + 2 = 2sqrt(2)
r(sqrt(2) +1) = 2sqrt(2) - 2
r=(2sqrt(2) - 2)/(sqrt(2) + 1)
Simplifying this gives us
r= 6 - 4sqrt(2)
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