1. **State the problem:** Points P, Q, R, and S are collinear with Q between P and R, and R between Q and S. Given that $PQ = RS$, $PS = 27$, and $PR = 22$, find the length $QR$. 2. **Set up variables and relationships:** Let $PQ = x$. Since $PQ = RS$, then $RS = x$. 3. **Express known lengths in terms of $x$ and $QR$:** - $PR = PQ + QR = x + QR = 22$ - $PS = PQ + QR + RS = x + QR + x = 2x + QR = 27$ 4. **Use the equations to solve for $x$ and $QR$:** From $PR = 22$, we have: $$x + QR = 22$$ From $PS = 27$, we have: $$2x + QR = 27$$ 5. **Subtract the first equation from the second:** $$ (2x + QR) - (x + QR) = 27 - 22 $$ $$ 2x + QR - x - QR = 5 $$ $$ x = 5 $$ 6. **Substitute $x = 5$ back into $x + QR = 22$:** $$5 + QR = 22$$ $$QR = 22 - 5 = 17$$ 7. **Final answer:** $$\boxed{17}$$ Thus, the length $QR$ is 17.