1. **Problem statement:** Given triangle ABC with segment BD bisecting angle ABC, sides AB = 18, BC = 19, AD = x, and DC = 11. We need to find the value of $x$. 2. **Formula used:** The Angle Bisector Theorem states that the bisector of an angle in a triangle divides the opposite side into segments proportional to the adjacent sides. Mathematically, $$\frac{AD}{DC} = \frac{AB}{BC}$$ 3. **Apply the theorem:** Substitute the known values: $$\frac{x}{11} = \frac{18}{19}$$ 4. **Solve for $x$:** Multiply both sides by 11: $$x = 11 \times \frac{18}{19}$$ 5. **Simplify:** $$x = \frac{11 \times 18}{19} = \frac{198}{19}$$ 6. **Calculate the decimal value:** $$x \approx 10.4211$$ 7. **Round to the nearest tenth:** $$x \approx 10.4$$ **Final answer:** $x = 10.4$