If largest circle has radius 225 and medium circle has radius 100 in the image, what is the radius of smallest circle?

Question

Three differently sized circles and a line can be positioned so that they are all mutually tangent to one another, as shown below. If the largest circle has radius 225 and the medium circle has radius 100, what is the radius of the smallest circle?

Answer 1

The answer is:

36

EXPLANATION:

The specialized case for Descartes' Theorem, k4 = k1 + k2 ± 2√(k1k2), where k is defined as the curvature 1/r for each circle.
In this case,
k4 = 1/225 + 1/100 + 2√((1/225)(1/100)) = 1/36, and so r = 36.

Answer 2

r must be 36