1. **Problem statement:** We are given a triangle with points A, B, C, and a point D on segment BC. We know $\angle DAC = \angle BAD$ and lengths $AC = 5.1$, $AB = 5.7$, and $BD = 3.6$. We need to find the length of $\overline{CD}$ rounded to one decimal place. 2. **Key insight:** Since $\angle DAC = \angle BAD$, point D lies on BC such that $\triangle DAC$ and $\triangle BAD$ share the same angle at A. This implies that $AD$ bisects $\angle BAC$. 3. **Angle bisector theorem:** The angle bisector theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides: $$\frac{BD}{DC} = \frac{AB}{AC}$$ 4. **Substitute known values:** $$\frac{3.6}{DC} = \frac{5.7}{5.1}$$ 5. **Solve for $DC$:** $$DC = \frac{3.6 \times 5.1}{5.7}$$ 6. **Calculate:** $$DC = \frac{18.36}{5.7} \approx 3.2211$$ 7. **Round to one decimal place:** $$DC \approx 3.2$$ **Final answer:** The length of $\overline{CD}$ is approximately **3.2**.