1. **State the problem:** Given triangle $\triangle QRS$ with circumcenter $W$, and segments $QR=32$, $RU=19$, $QV=24$, and $VS=21$, find the lengths of $RS$, $TQ$, $WS$, $QV$, and $TW$, rounding to the tenths place. 2. **Recall properties:** The circumcenter $W$ is equidistant from all vertices $Q$, $R$, and $S$. Points $U$, $V$, and $T$ are midpoints of sides $RS$, $QS$, and $QR$ respectively. 3. **Find $RS$:** Since $U$ is midpoint of $RS$, $RU=19$ means $RS=2 \times RU=2 \times 19=38$. 4. **Find $TQ$:** $T$ is midpoint of $QR$, so $QT=TR=\frac{QR}{2}=\frac{32}{2}=16$. 5. **Find $WS$:** Since $W$ is circumcenter, $WQ=WR=WS$. We need to find $WQ$ or $WR$. 6. **Find $QV$:** Given $QV=24$ (already provided). 7. **Find $TW$:** $T$ is midpoint of $QR$, and $W$ is circumcenter. Since $W$ is equidistant from $Q$ and $R$, $W$ lies on the perpendicular bisector of $QR$ passing through $T$. The distance $TW$ is the distance from midpoint $T$ to $W$. 8. **Calculate $WQ$:** Using the triangle and midpoint properties, $V$ is midpoint of $QS$, so $QV=VS=24$ and $21$ given, but $VS=21$ is given, so $QS=QV+VS=24+21=45$. 9. **Since $W$ is circumcenter, $WQ=WR=WS=r$ (circumradius). We can find $r$ using the right triangle formed by $W$, $T$, and $Q$ or $R$. 10. **Calculate $TW$:** Since $T$ is midpoint of $QR$, $QT=16$. The circumcenter lies on the perpendicular bisector of $QR$, so $TW$ is the height from $W$ to $QR$. 11. **Without coordinates, approximate $WS$ as equal to $WQ$ or $WR$. Since $W$ is equidistant, and $WQ$ is unknown, we cannot find exact $WS$ or $TW$ without coordinates or more info. **Final answers:** - $RS=38.0$ - $TQ=16.0$ - $WS=\text{unknown without more info}$ - $QV=24.0$ - $TW=\text{unknown without more info}$