1. **State the problem:** We are given triangle PNO with points Q on PN and R on PO such that QR is parallel to NO. Given lengths are RO = 12.7, PQ = 11.5, and PR = 15.3. We need to find the length of QN. 2. **Identify the theorem:** Since QR is parallel to NO, by the Basic Proportionality Theorem (Thales' theorem), the segments on PN and PO are divided proportionally: $$\frac{PQ}{QN} = \frac{PR}{RO}$$ 3. **Write the known values:** $$PQ = 11.5, \quad PR = 15.3, \quad RO = 12.7$$ 4. **Set up the proportion:** $$\frac{11.5}{QN} = \frac{15.3}{12.7}$$ 5. **Solve for $QN$:** Multiply both sides by $QN$: $$11.5 = QN \times \frac{15.3}{12.7}$$ Divide both sides by $\frac{15.3}{12.7}$: $$QN = \frac{11.5}{\frac{15.3}{12.7}}$$ Show cancellation: $$QN = 11.5 \times \frac{12.7}{\cancel{15.3}} \times \frac{\cancel{1}}{1}$$ Calculate: $$QN = 11.5 \times \frac{12.7}{15.3} = \frac{11.5 \times 12.7}{15.3}$$ Calculate numerator: $$11.5 \times 12.7 = 146.05$$ Divide: $$QN = \frac{146.05}{15.3} \approx 9.55$$ 6. **Round the answer:** Rounded to the nearest tenth: $$QN \approx 9.6$$ **Final answer:** The length of $QN$ is approximately 9.6.