Let's analyze and verify the ratios of the widths and lengths in the diagram.

Step 1: Ratio of the Widths

In the smaller shape (on the left), the width is given as 6. In the larger shape (on the right), the width is given as 18.

The ratio of the widths is: $\text{Width ratio} = \frac{\text{width of larger shape}}{\text{width of smaller shape}} = \frac{18}{6} = 3$

Step 2: Ratio of the Lengths

In the smaller shape, the height (length) is given as 8. In the larger shape, the height is given as 24.

The ratio of the lengths is: $\text{Length ratio} = \frac{\text{length of larger shape}}{\text{length of smaller shape}} = \frac{24}{8} = 3$

Conclusion:

The ratio of the widths (18:6) and the ratio of the lengths (24:8) are both equal to 3. Therefore, Casey's ratios are equal, and her reasoning that the zoom factor multiplies each part of the original shape consistently is verified.

Do you have any questions or would you like further clarification?

Here are 5 related questions to expand your understanding:

1. How would the area change if the zoom factor is applied uniformly?
2. Can you calculate the area of both shapes and compare them?
3. What happens to the perimeter of the shapes when you apply the zoom factor?
4. How would the ratios change if the zoom factor were different, say 4 instead of 3?
5. Can this concept of ratio and zoom factor be applied to 3D shapes like cubes or spheres?

Tip: When verifying ratios or proportions in similar figures, remember that all linear dimensions (length, width, height) scale by the same factor, but areas scale by the square of that factor.