1. **Problem statement:** We are given two triangles, $\triangle ARS$ and $\triangle EST$, sharing point $S$. Segments $\overline{RA}$ and $\overline{ET}$ are parallel. We know $RT = 21$ units, $RA = 6$ units, and $ET = 8$ units. We need to find the length of $\overline{ST}$. 2. **Key concept:** Since $\overline{RA} \parallel \overline{ET}$, triangles $ARS$ and $EST$ are similar by the AA (Angle-Angle) similarity criterion. 3. **Similarity ratio:** Corresponding sides of similar triangles are proportional. So, $$\frac{RA}{ET} = \frac{RS}{ES} = \frac{RT}{ST}$$ 4. **Known values:** $$\frac{RA}{ET} = \frac{6}{8} = \frac{3}{4}$$ 5. **Using the ratio for $RT$ and $ST$:** $$\frac{RT}{ST} = \frac{3}{4}$$ 6. **Substitute $RT = 21$:** $$\frac{21}{ST} = \frac{3}{4}$$ 7. **Solve for $ST$:** Multiply both sides by $ST$ and then by $\frac{4}{3}$: $$21 = \frac{3}{4} ST$$ $$\Rightarrow ST = 21 \times \frac{4}{3}$$ 8. **Simplify:** $$ST = 21 \times \frac{4}{3} = \cancel{21} \times \frac{4}{\cancel{3}} = 7 \times 4 = 28$$ 9. **Final answer:** $$ST = 28$$ units. Thus, the length of $\overline{ST}$ is 28 units.