Similar Triangles

$\overline{TR}=20*\frac{2}{5+2}=\frac{40}{7}$.

Set the value of $\overline{UV}=x$.

 $\because UVQP\sim TRVU$.

$\therefore \frac{TR}{UV}=\frac{UV}{PQ}$

$\therefore \frac{\frac{40}{7}}{x}=\frac{x}{20}$

${x}^{2}=\frac{800}{7}$

$x=\frac{\sqrt{800}}{\sqrt{7}}=\frac{20\sqrt{14}}{7}$