https://file.aiquickdraw.com/mathbot/file/1726706569557u5unlyox.jpg

To determine the scale factor for the dilation, we need to compare the distances between corresponding points before and after the dilation.

From the image:
- \( L \) is mapped to \( L' \),
- \( M \) is mapped to \( M' \),
- \( N \) is mapped to \( N' \).

Let's calculate the distance between a pair of corresponding points. We will calculate the distance between \( M \) and \( M' \) before and after the dilation.

### Coordinates of the points:
- \( M = (3, 4) \)
- \( M' = (6, 8) \)

We can calculate the scale factor by finding the ratio of the distances from the origin to the original point \( M \) and the dilated point \( M' \).

1. Distance from the origin to \( M \):
   \[
   d(M) = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
   \]

2. Distance from the origin to \( M' \):
   \[
   d(M') = \sqrt{(6 - 0)^2 + (8 - 0)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
   \]

The scale factor \( k \) is the ratio of the distance from the origin to \( M' \) over the distance to \( M \):
\[
k = \frac{d(M')}{d(M)} = \frac{10}{5} = 2
\]

### Conclusion:
The scale factor of the dilation is **2**.

Do you want more details or have any questions?

Here are some related questions:
1. How do you verify if the dilation is centered at the origin?
2. Can a dilation have a fractional scale factor?
3. How do you find the scale factor if the dilation is not centered at the origin?
4. What happens to angles during a dilation?
5. How does dilation affect the area of a shape?

**Tip**: In dilation, the corresponding sides of the preimage and image are proportional by the scale factor, but angles remain unchanged.

Find the scale factor that was used in the dilation:

This problem demonstrates how to calculate the scale factor for a dilation using points on a coordinate grid. Learn how to apply the distance formula and understand the concept of transformations in geometry. Suitable for students in Grades 8-10.

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