1. **State the problem:** We have trapezoid PQST with \overline{ST} parallel to \overline{PQ}. Given lengths are $RQ=36$, $RT=22.5$, $PR=24$, and $PQ=32$. We need to find the length of \overline{ST}. 2. **Recall the property of trapezoids with intersecting diagonals:** The diagonals intersect at point R, and the segments satisfy the ratio $\frac{PR}{RQ} = \frac{RT}{TS}$. This is because the diagonals of a trapezoid are divided proportionally by their intersection point. 3. **Set up the proportion:** Using the given lengths, $\frac{PR}{RQ} = \frac{RT}{TS} \Rightarrow \frac{24}{36} = \frac{22.5}{TS}$. 4. **Simplify the left side:** $\frac{24}{36} = \frac{2}{3}$. So, $\frac{2}{3} = \frac{22.5}{TS}$. 5. **Solve for $TS$:** Cross-multiply: $2 \times TS = 3 \times 22.5$ $$2 \times TS = 67.5$$ Divide both sides by 2: $$\cancel{2} \times TS = \frac{67.5}{\cancel{2}}$$ $$TS = 33.75$$ 6. **Find $ST$:** Since $ST = TS$, the length of \overline{ST} is $33.75$. 7. **Summary:** The length of the top base \overline{ST} in trapezoid PQST is $33.75$.