Question: Points $P$, $Q$, $R$, and $S$ are collinear. Point $Q$ is between $P$ and $R$, $R$ is between $Q$ and $S$, and $PQ \cong RS$. If $PS = 22$ and $PR = 19$, what is the value of $QR$?

1. **State the problem:** We have points $P$, $Q$, $R$, and $S$ on a line in that order. Given that $Q$ is between $P$ and $R$, $R$ is between $Q$ and $S$, and $PQ \cong RS$. Also, $PS = 22$ and $PR = 19$. We need to find the length $QR$. 2. **Set variables and known lengths:** - Let $PQ = x$. - Since $PQ \cong RS$, then $RS = x$. - We know $PR = 19$ and $PS = 22$. 3. **Express lengths in terms of $x$ and $QR$:** - Since $Q$ is between $P$ and $R$, $PR = PQ + QR = x + QR = 19$. - Since $R$ is between $Q$ and $S$, $PS = PQ + QR + RS = x + QR + x = 2x + QR = 22$. 4. **Write the system of equations:** $$\begin{cases} x + QR = 19 \\ 2x + QR = 22 \end{cases}$$ 5. **Subtract the first equation from the second:** $$ (2x + QR) - (x + QR) = 22 - 19 $$ $$ 2x + QR - x - QR = 3 $$ $$ x = 3 $$ 6. **Find $QR$ using $x + QR = 19$:** $$ 3 + QR = 19 $$ $$ QR = 19 - 3 = 16 $$ 7. **Final answer:** $$ \boxed{16} $$ Thus, the length $QR$ is $16$.