Geometry

$\angle{P} = \angle{TUV}$

$\angle{PUV} = \angle{UTR}$

$\angle{PQV} = \angle{UVR}$

$\angle{QVU} = \angle{VRT}$

Therefore, trapezoid PUVQ is similar to trapezoid UTRV

$\overline{TR} = \frac{2}{7}(20) = \frac{40}{7}$

Since $PUVQ \sim UTRV$:

$\frac{20}{UV} = \frac{UV}{\frac{40}{7}}$

$\frac{800}{7} = \overline{UV}^2$

$\textbf{UV = }\frac{\textbf{20}\sqrt{\textbf{2}}}{\sqrt{\textbf{7}}}$

$\textbf{UV = }\frac{\textbf{20}\sqrt{\textbf{14}}}{{\textbf{7}}}$

Let me know if I made any mistakes!