1. **Stating the problem:** We are given a triangle ABC with points D and E on sides AC and AB respectively, and segments BD, BE, and EA inside the triangle. The side lengths AB=11 and BC=8 are given, along with fractions 21/4, 32/6, 54/5, and 16/6 which likely represent segment lengths or ratios. 2. **Understanding the problem:** We need to analyze the segments BD, BE, EA, and BC, possibly to find relationships or lengths using the given fractions and side lengths. 3. **Interpreting the fractions:** Simplify the fractions: - $\frac{21}{4} = 5.25$ - $\frac{32}{6} = \frac{16}{3} \approx 5.33$ - $\frac{54}{5} = 10.8$ - $\frac{16}{6} = \frac{8}{3} \approx 2.67$ 4. **Using triangle properties:** Since D and E lie on sides AC and AB, and BD is drawn, we can use the triangle side lengths and segment ratios to find unknown lengths or verify segment relationships. 5. **Example calculation:** If BE and EA are parts of AB=11, and BE + EA = AB, then check if $5.33 + 2.67 = 8$ which is less than 11, so these might be partial lengths or ratios. 6. **Conclusion:** Without additional context or specific questions, the problem involves understanding segment lengths and ratios inside triangle ABC with given side lengths and fractions representing segment lengths. Final answer: The simplified fractions are $5.25$, $5.33$, $10.8$, and $2.67$. The side lengths AB=11 and BC=8 are given. Further calculations require more specific questions or instructions.