1. **Stating the problem:** We have a triangle with points T, P, and R. Inside the triangle, segment QS is parallel to PT. Given lengths are TS = 42, SR = 21, and RQ = 9. We need to find the length PR. 2. **Understanding the problem:** Since QS is parallel to PT, triangles TSR and QSR are similar by the Basic Proportionality Theorem (Thales' theorem). 3. **Using the theorem:** The ratio of the segments on one side equals the ratio on the other side: $$\frac{TS}{SR} = \frac{RQ}{PR}$$ 4. **Substitute known values:** $$\frac{42}{21} = \frac{9}{PR}$$ 5. **Simplify the left side:** $$\frac{\cancel{42}}{\cancel{21}} = 2 = \frac{9}{PR}$$ 6. **Solve for PR:** $$2 = \frac{9}{PR} \implies 2 \times PR = 9 \implies PR = \frac{9}{2} = 4.5$$ **Final answer:** $$PR = 4.5$$