We are given that:

• Point $M$ is the midpoint of line segment $AB$,
• Point $N$ is the midpoint of line segment $AM$,
• The length of segment $AN = 7$.

Step-by-step solution:

1. Let the total length of line segment $AB$ be $L$.

2. Since $M$ is the midpoint of $AB$, the length of segment $AM$ is half of $AB$: $AM = \frac{L}{2}$

3. Point $N$ is the midpoint of $AM$, so the length of segment $AN$ is half of $AM$: $AN = \frac{AM}{2} = \frac{\frac{L}{2}}{2} = \frac{L}{4}$

4. We are told that $AN = 7$, so: $\frac{L}{4} = 7$

5. Solving for $L$: $L = 7 \times 4 = 28$

Conclusion:

The length of line segment $AB$ is $\boxed{28}$.

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:

1. What is the formula to find the midpoint of a line segment on a coordinate plane?
2. How do you find the length of a line segment given its endpoints in a coordinate plane?
3. If the midpoint is given, how do you find the endpoints of a segment?
4. How can you divide a line segment into any ratio?
5. What are some real-world applications of midpoints?

Tip: Midpoints divide a line segment into two equal parts, which is crucial for geometric constructions and problem-solving.