1. **Problem Statement:** We are given a triangle with points A, B, C, and D inside it. We know that $\angle DAC = \angle BAD$ and the lengths $AC = 4.6$, $CD = 2.5$, and $AB = 6.8$. We need to find the length of $\overline{BD}$ rounded to one decimal place. 2. **Key Insight:** Since $\angle DAC = \angle BAD$, point D lies on the angle bisector of $\angle BAC$. By the Angle Bisector Theorem, the angle bisector divides the opposite side into segments proportional to the adjacent sides. 3. **Angle Bisector Theorem:** If $D$ lies on $BC$ such that $AD$ bisects $\angle BAC$, then $$\frac{BD}{DC} = \frac{AB}{AC}$$ 4. **Apply the theorem:** Substitute the known values: $$\frac{BD}{2.5} = \frac{6.8}{4.6}$$ 5. **Solve for $BD$:** $$BD = 2.5 \times \frac{6.8}{4.6}$$ Calculate the fraction: $$\frac{6.8}{4.6} \approx 1.4783$$ Multiply: $$BD \approx 2.5 \times 1.4783 = 3.6958$$ 6. **Round the answer:** $$BD \approx 3.7$$ **Final answer:** The length of $\overline{BD}$ is approximately **3.7** units.