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.