1. The problem involves finding the length of segment $RT$ in a geometric figure with points $T$, $R$, $S$, and $U$. 2. Given that $TU$ and $US$ are perpendicular, and $UR = 8$, and $TS = RS$ (indicating $\triangle TSR$ is isosceles). 3. Since $TS = RS$, $\triangle TSR$ is isosceles with $T$ and $R$ as vertices and $S$ as the apex. 4. The right angle at $U$ suggests $\triangle TUS$ and $\triangle USR$ are right triangles. 5. To find $RT$, we can use the Pythagorean theorem or properties of isosceles triangles depending on additional lengths or angles. 6. However, with the given information, if $UR = 8$ and $TS = RS$, and $U$ is the foot of the perpendicular from $T$ or $R$ to $US$, then $RT$ is the hypotenuse of the right triangle with legs $UR = 8$ and $TU$ or $US$. 7. Without the length of $TU$ or $US$, we cannot numerically calculate $RT$. 8. If $TU = US$, then $RT = 8\sqrt{2}$ by the Pythagorean theorem. Final answer: $RT = 8\sqrt{2}$ assuming $TU = US$ and right angle at $U$.