1. **Problem statement:** We are given triangle ABC with angle bisector AD, where $\angle DAC = \angle BAD$. We need to find the length of segment $\overline{BD}$ given $AC = 5.8$, $CD = 2.5$, and $AB = 6.2$.\n\n2. **Key formula:** The Angle Bisector Theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. Specifically, $\frac{BD}{DC} = \frac{AB}{AC}$.\n\n3. **Apply the theorem:** Substitute the known values:\n$$\frac{BD}{2.5} = \frac{6.2}{5.8}$$\n\n4. **Solve for $BD$:** Multiply both sides by 2.5:\n$$BD = 2.5 \times \frac{6.2}{5.8}$$\n\n5. **Simplify the fraction:**\n$$BD = 2.5 \times \frac{6.2}{5.8} = 2.5 \times 1.0689655...$$\n\n6. **Calculate the product:**\n$$BD \approx 2.5 \times 1.069 = 2.6725$$\n\n7. **Round to one decimal place:**\n$$BD \approx 2.7$$\n\n**Final answer:** The length of $\overline{BD}$ is approximately $2.7$ units.