PQRS and QTUV are squares

𝑅𝑋 = \(\frac{1}{2}\) = \(SX \)

So \(PQRS\) is a square with a side length of \(1\)

Right triangle SPW, \(SP^2 + PW^2 = SW^2\)

\(SW^2 = 1 + \frac{1}{4} = \frac{5}{4}\)

\(SW\) ​= \(\frac{\sqrt{5}}{2}\) = \(TW\)

\(TQ = TW - QW\)

\(QW = \frac{1}{2}\)

So TQ = \(\frac{\sqrt{5}}{2}\) - \(\frac{1}{2}\)

\(\overline {TQ}\) = \(\overline {UV}\) = \(\frac{\sqrt{5}-1}{2}\)

Answer A