What is the length of the marked segment?



The marked segment is the hypotenuse of a right triangle such that one leg is a height of the big equilateral triangle.

The other leg has length \[\dfrac{2}{3}-\dfrac{1}{2}=\dfrac{1}{6}\,.\] By Pythagoras's theorem the marked segment has length \[\sqrt{\Big(\dfrac{1}{6}\Big)^2+\Big(\dfrac{\sqrt{3}}{2}\Big)^2}=\dfrac{\sqrt{7}}{3}\,.\]