1. **Problem Statement:** Find the length of $\overline{BL}$ given that $\overline{TS}$ is parallel to $\overline{AL}$, with $AL=12$, $TS=7.75$, and $SL=5.1$. 2. **Key Concept:** When two segments are parallel, corresponding triangles formed are similar. Here, triangles involving $AL$ and $TS$ are similar. 3. **Using Similar Triangles:** Since $\overline{TS} \parallel \overline{AL}$, triangles $BTS$ and $BAL$ are similar. 4. **Set up the ratio of corresponding sides:** $$\frac{TS}{AL} = \frac{SL}{BL}$$ 5. **Substitute known values:** $$\frac{7.75}{12} = \frac{5.1}{BL}$$ 6. **Solve for $BL$:** Multiply both sides by $BL$: $$BL \times \frac{7.75}{12} = 5.1$$ Divide both sides by $\frac{7.75}{12}$: $$BL = \frac{5.1}{\frac{7.75}{12}}$$ Show cancellation: $$BL = 5.1 \times \frac{12}{\cancel{7.75}} \times \frac{\cancel{7.75}}{12} = 5.1 \times \frac{12}{7.75}$$ 7. **Calculate:** $$BL = \frac{5.1 \times 12}{7.75} = \frac{61.2}{7.75} \approx 7.9$$ 8. **Final answer:** $$BL \approx 7.90 \text{ units}$$