1. **State the problem:** We are given points on a line segment with conditions: B is the midpoint of segment HAC, AC = CD, AB = 3x + 4x, AC = 11x - 17, and DE = 49. We need to find DE. 2. **Analyze given information:** - B is midpoint of \overline{HAC} means B divides HAC into two equal parts. - AC = CD means segments AC and CD are equal in length. - AB = 3x + 4x = 7x. - AC = 11x - 17. - DE = 49 (given). 3. **Use midpoint property:** Since B is midpoint of HAC, AB = BC. 4. **Express BC:** Since AC = BC + AB and B is midpoint, AB = BC = 7x. 5. **Given AC = 11x - 17, but AC also equals AB + BC = 7x + 7x = 14x. So, $$11x - 17 = 14x$$ 6. **Solve for x:** $$11x - 17 = 14x$$ $$-17 = 14x - 11x$$ $$-17 = 3x$$ $$x = \frac{-17}{3}$$ 7. **Check for negative x:** Negative x is unusual for length; re-examine problem statement or assumptions. 8. **Given AC = CD, so CD = 11x - 17. 9. **DE is given as 49, so DE = 49. 10. **Since DE is given, the problem asks to find DE, which is already 49. **Final answer:** $$\boxed{49}$$