What is the length of the marked segment?



The quadrilateral \(ABCD\) is a square, see .
The segment \(BD\) is the diagonal of that square.
The side length of \(ABCD\) is \(\sqrt{2-\sqrt{3}}\) ,see .
So the requested length is \[\sqrt{2}\cdot \sqrt{2-\sqrt{3}}\,.\]