1. **Problem statement:** We have triangle ABC with point D on side AC such that segment BD bisects angle ABC. Given AB = 19, AD = 14, DC = 17, and BC = x, we need to find the value of x. 2. **Formula used:** The Angle Bisector Theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. Mathematically, $ \frac{AB}{BC} = \frac{AD}{DC} $. 3. **Apply the theorem:** Substitute the known values: $$\frac{19}{x} = \frac{14}{17}$$ 4. **Solve for x:** Cross-multiply: $$19 \times 17 = 14 \times x$$ $$323 = 14x$$ 5. **Isolate x:** $$x = \frac{323}{14}$$ 6. **Simplify and calculate:** $$x = \frac{\cancel{323}}{\cancel{14}} = 23.0714...$$ 7. **Round to the nearest tenth:** $$x \approx 23.1$$ **Final answer:** $$x = 23.1$$