The problem asks for the length of the line segment $AB$ in the given figure, where there are a series of connected right triangles, each with leg lengths of $1$ meter.

Solution:

1. Since each triangle in the figure is a right triangle with one leg of $1$ meter, we can use the Pythagorean theorem to calculate the hypotenuse of each triangle.

2. For each right triangle, with legs of $1$ meter and $1$ meter: $\text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2}$

3. Moving from point $B$ to point $A$, there are three right triangles that are connected. Each hypotenuse contributes a length of $\sqrt{2}$ meters along the segment $AB$.

4. Therefore, the total length of $AB$ is: $AB = 3 \times \sqrt{2} = 3\sqrt{2} \text{ meters}$

Final Answer:

$AB = 3\sqrt{2} \text{ meters}$

Would you like a more detailed explanation of the steps or have any questions?

Related Questions:

1. How do you calculate the hypotenuse of a right triangle with legs of different lengths?
2. What is the Pythagorean theorem, and how is it used in geometry?
3. If each triangle had legs of $2$ meters, how would the length of $AB$ change?
4. How can the length $3\sqrt{2}$ be approximated as a decimal?
5. How do geometric series apply to similar right triangles?

Tip:

When dealing with right triangles, always check if you can simplify the expression using the Pythagorean theorem, especially in figures with repeated geometric patterns like this one.