Answer: D

Small Hint:

For two circles of radii $r$ and $s$ both tangent to a line and to each other, their contact points on the line are $2\sqrt{rs}$ apart

Big Hint:

The contact point of the small circle splits the segment: $2\sqrt{x} + 2\sqrt{4x} = 2\sqrt{4}$

Solution:

When two mutually tangent circles of radii $r$ and $s$ both rest on a line, the distance between their points of tangency is $2\sqrt{rs}.$

The big circles’ contact points are $2\sqrt{1 \cdot 4} = 4$ apart. Placing the small circle of radius $x$ between them, its two tangent distances add up: $$2\sqrt{1 \cdot x} + 2\sqrt{4 \cdot x} = 4.$$

Then $6\sqrt{x} = 4,$ so $\sqrt{x} = \dfrac{2}{3}$ and $x = \dfrac{4}{9}.$

Thus, the correct answer is D.