1. **Problem statement:** We have a quadrilateral subdivided into two triangles by a segment DE parallel to AB. Given lengths are $AB=8$, $DE=5$, and $BC=10$. We need to find the length $BE=x$. 2. **Key concept:** Since $DE \parallel AB$, triangles $ADE$ and $ABC$ are similar by the AA similarity criterion. 3. **Similarity ratio:** The ratio of corresponding sides in similar triangles is equal. So, $$\frac{DE}{AB} = \frac{BE}{BC}$$ 4. **Substitute known values:** $$\frac{5}{8} = \frac{x}{10}$$ 5. **Solve for $x$:** Multiply both sides by 10: $$10 \times \frac{5}{8} = x$$ 6. **Simplify:** $$x = \frac{10 \times 5}{8} = \frac{50}{8}$$ 7. **Reduce fraction:** $$x = \frac{\cancel{50}^{\times 2 \times 25}}{\cancel{8}^{\times 2 \times 4}} = \frac{25}{4} = 6.25$$ **Final answer:** $$x = 6.25$$