What is the length of the marked segment?



Let \(a\) be the length of the marked segment, which is the side length of the square and also of the equilateral triangle, and let \(h\) be the height of the equilateral triangle.
We have the following system

\(\begin{cases} 2h+a = 1 \quad \quad (1)\\ h=\dfrac{\sqrt{3}a}{2} \quad \quad (2) \end{cases}\)

This gives us that \begin{align*} &2\cdot\dfrac{\sqrt{3}a}{2} + a = 1\\[5pt] \implies &\sqrt{3}a + a = 1\\ \implies&(\sqrt{3} + 1) a = 1\\[5pt] \implies& a = \dfrac{1}{\sqrt{3} + 1}\\[5pt] \implies& a = \dfrac{\sqrt{3}-1}{2} \end{align*} Hence the marked segment has length \(\dfrac{\sqrt{3}-1}{2}\).