In the figure above, there are 4 circles that are tangent to one another. The two largest circles have radius 3 and the second in size has radius 2. Find the radius of the small circle.
Let r be the radius of the small circle.
\(\sqrt{(3 + r)^2 - 3^2} + r + 2 = \sqrt{5^2 - 3^2}\\ \sqrt{r(6 + r)} + r + 2 = 4\\ \sqrt{r(6 + r)} = 2 - r\\ r^2 + 6r = r^2 - 4r + 4\\ 10r = 4\\ r = \dfrac{2}{5}\)
