1. **Problem statement:** Given points P, Q, R on a slanted line and points U, T, S on a slanted baseline, with vertical lines through P-U, Q-T, and R-S, find the length of segment PR. 2. **Given data:** - QR = 32 - UT = 21 - TS = 28 3. **Understanding the problem:** Since P, Q, R are on the same slanted line and U, T, S are on the baseline with vertical lines connecting them, triangles formed are similar by AA similarity (corresponding angles are equal). 4. **Using similarity:** The segments on the baseline (UT and TS) correspond to segments on the top line (QR and PR). Since vertical lines are parallel, the ratios of corresponding segments are equal: $$\frac{PR}{QR} = \frac{UT + TS}{UT}$$ 5. **Calculate the ratio:** $$UT + TS = 21 + 28 = 49$$ 6. **Set up the proportion:** $$\frac{PR}{32} = \frac{49}{21}$$ 7. **Solve for PR:** $$PR = 32 \times \frac{49}{21}$$ 8. **Simplify the fraction:** $$\frac{49}{21} = \frac{\cancel{7} \times 7}{\cancel{7} \times 3} = \frac{7}{3}$$ 9. **Calculate PR:** $$PR = 32 \times \frac{7}{3} = \frac{32 \times 7}{3} = \frac{224}{3} \approx 74.67$$ **Final answer:** $$PR = \frac{224}{3} \text{ or approximately } 74.67$$