To find the length of line segment $AB$, we need to analyze the diagram with the three circles.

• The diameters of the circles are given as:
  • Smallest circle: $3 \, \text{cm}$ (radius $= 1.5 \, \text{cm}$),
  • Middle circle: $4 \, \text{cm}$ (radius $= 2 \, \text{cm}$),
  • Largest circle: $6 \, \text{cm}$ (radius $= 3 \, \text{cm}$).

Conceptual Approach:

The line segment $AB$ is the vertical line passing through the centers of the three circles. The length of $AB$ is essentially the sum of the radii of all three circles because the centers of the circles lie along the same vertical line.

• The total length of $AB$ can be calculated as: $AB = \text{radius of smallest circle} + \text{radius of middle circle} + \text{radius of largest circle}$ $AB = 1.5 \, \text{cm} + 2 \, \text{cm} + 3 \, \text{cm} = 6.5 \, \text{cm}$

Thus, the length of $AB$ is 6.5 cm.

Would you like any further details or explanations on this?

Here are some related questions for practice:

1. How would the length of $AB$ change if the radius of one of the circles increased?
2. What if the circles were not touching each other? How would we calculate $AB$?
3. Can you find the area of each circle based on the given diameters?
4. What if the arrangement of circles was horizontal instead of vertical? How would that affect the result?
5. How can you calculate the perimeter of the rectangle enclosing the circles?

Tip: In problems like these, focusing on symmetry and how shapes are arranged often simplifies complex-looking diagrams!