What is the length of the marked segment?



The triangle \(ABC\) is a right triangle in \(A\), because it is the adjacent angle to a right angle , see or .
The angle in \(C\) equals \(30^\circ\), because it is half the angle of an equilateral triangle.
The length of \(AC\) is \(\dfrac{2-\sqrt{3}}{2}\), see .
So we obtain the length of the marked segment \(AB\) by \[AB = \tan 30^\circ \cdot \dfrac{2-\sqrt{3}}{2} = \dfrac{2\sqrt{3}-3}{6}\,.\]