Point A, B, and C all lie on a line. Point D is midpoint of `bar(AB)` and E is the midpoint of `BC, bar(AB)=4 and bar(BC)=10`. Which of the following could be the length of `bar(AE)`?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to find the possible lengths of segment \( \bar{AE} \) given that points A, B, and C lie on a line, with D as the midpoint of \( \bar{AB} \) and E as the midpoint of \( \bar{BC} \). The lengths of \( \bar{AB} \) and \( \bar{BC} \) are given as 4 and 10, respectively.

### Step-by-Step Solution:

1. **Define Points on a Number Line**:
- Let point A be at position 0 on the number line.
- Since \( \bar{AB} = 4 \), point B will be at position 4 (i.e., \( B = A + 4 = 0 + 4 = 4 \)).
- Now, we need to determine the position of point C. Since \( \bar{BC} = 10 \), point C can be either to the right or left of point B.

2. **Case 1: C is to the Right of B**:
- If C is to the right of B, then \( C = B + 10 = 4 + 10 = 14 \).
- Now, we find the midpoint E of \( \bar{BC} \):
\[ E = \frac{B + C}{2} = \frac{4 + 14}{2} = \frac{18}{2} = 9 \]
- Now, we calculate the length of \( \bar{AE} \):
\[ \bar{AE} = E - A = 9 - 0 = 9 \]

3. **Case 2: C is to the Left of B**:
- If C is to the left of B, then \( C = B - 10 = 4 - 10 = -6 \).
- Again, we find the midpoint E of \( \bar{BC} \):
\[ E = \frac{B + C}{2} = \frac{4 + (-6)}{2} = \frac{-2}{2} = -1 \]
- Now, we calculate the length of \( \bar{AE} \):
\[ \bar{AE} = E - A = -1 - 0 = -1 \quad (\text{but we take absolute value, so } 1) \]

4. **Conclusion**:
- From both cases, we find that the possible lengths of \( \bar{AE} \) are 9 (from Case 1) and 1 (from Case 2).
- Since the question asks for which of the following could be the length of \( \bar{AE} \), we see that 1 is a valid option.

### Final Answer:
The possible length of \( \bar{AE} \) is **1**.