Consider the condition: what is the length of the common tangent when two circles of radii r1 and r2 touch externally?
Here AB (the common tangent)
= O′C = \(\sqrt{OO'^2 -OC^2}\)
= \(\sqrt{(r_1+r_2)^2 - (r_1 -r_2)^2}\)
= \(\sqrt {4r_1\,r_2}\) = \(2\sqrt{r_1\,r_2}\)
Therefore, according to the given figure, PR is the length of the common tangent to circle of radii r and 4.
∴ PQ = \(2\sqrt{4r}\) = \(4\sqrt{r}\)
QR = \(2\sqrt{4r}\) = \(4\sqrt{r}\)
∵ PR = PQ + QR
∴ 2r = \(4\sqrt{r}\) + \(4\sqrt{r}\)
⇒ r = \(4\sqrt{r}\)
⇒ r2 = 16r
⇒ r = 16 cm.