1. **Problem statement:** Given triangle ABC with point D on side BC, angles $\angle DAC = \angle BAD$, and lengths $AC = 6.5$, $CD = 3.4$, and $AB = 4.9$, find the length of $BD$ rounded to one decimal place. 2. **Key insight:** Since $\angle DAC = \angle BAD$, point D lies on BC such that $AD$ bisects $\angle BAC$. By the Angle Bisector Theorem, the ratio of the two segments on BC created by D is equal to the ratio of the adjacent sides: $$\frac{BD}{DC} = \frac{AB}{AC}$$ 3. **Apply the theorem:** Substitute known values: $$\frac{BD}{3.4} = \frac{4.9}{6.5}$$ 4. **Solve for $BD$:** $$BD = 3.4 \times \frac{4.9}{6.5}$$ 5. **Calculate:** $$BD = 3.4 \times 0.7538 = 2.563$$ 6. **Round to one decimal place:** $$BD \approx 2.6$$ **Final answer:** The length of $BD$ is approximately **2.6**.