1. **Problem statement:** Given triangle OPQ with points R on OQ and S on PQ such that segment RS is parallel to segment OP. We know $OP=5.6$, $RQ=4.8$, and $OR=3.6$. We need to find the length of $RS$. 2. **Key concept:** When a segment is drawn parallel to one side of a triangle, it creates similar triangles. Here, triangle ORS is similar to triangle OPQ because $RS \parallel OP$. 3. **Similarity ratios:** Since $RS \parallel OP$, the sides are proportional: $$\frac{OR}{OQ} = \frac{RS}{OP}$$ 4. **Calculate $OQ$:** Since $R$ lies on $OQ$, and $OR=3.6$, $RQ=4.8$, then $$OQ = OR + RQ = 3.6 + 4.8 = 8.4$$ 5. **Set up proportion and solve for $RS$:** $$\frac{OR}{OQ} = \frac{RS}{OP} \implies \frac{3.6}{8.4} = \frac{RS}{5.6}$$ 6. **Cross multiply:** $$3.6 \times 5.6 = 8.4 \times RS$$ 7. **Calculate left side:** $$3.6 \times 5.6 = 20.16$$ 8. **Solve for $RS$:** $$RS = \frac{20.16}{8.4}$$ 9. **Simplify fraction:** $$RS = \frac{\cancel{20.16}}{\cancel{8.4}} = 2.4$$ **Final answer:** $$RS = 2.4$$