What is the length of the marked segment?





The marked segment is the leg of an isosceles triangle with height \(\frac{1}{2}\) and basis \(\sqrt{2}-1\), see .

By Pythagoras's theorem the length of the marked segment is \[\sqrt{\Big(\frac{1}{2}\Big)^2+\Big(\frac{\sqrt{2}-1}{2}\Big)^2}=\sqrt{1-\frac{\sqrt{2}}{2}}\,.\]