The problem asks to find the scale factor of the dilation between the rectangles $CDEF$ (larger rectangle) and $C'D'E'F'$ (smaller rectangle) on the coordinate plane.

To find the scale factor of dilation, we need to compare the lengths of corresponding sides of the two rectangles. Let's calculate the distances between two pairs of corresponding points.

1. Find the length of side $CD$ in the original rectangle $CDEF$:

• The coordinates of point $C$ are approximately $(-8, 4)$ and point $D$ is approximately $(-8, -4)$.
• The length of side $CD$ is the vertical distance between these points: $|4 - (-4)| = 4 + 4 = 8 \text{ units}.$

2. Find the length of side $C'D'$ in the dilated rectangle $C'D'E'F'$:

• The coordinates of point $C'$ are approximately $(-4, 2)$ and point $D'$ is approximately $(-4, -2)$.
• The length of side $C'D'$ is: $|2 - (-2)| = 2 + 2 = 4 \text{ units}.$

3. Calculate the scale factor:

The scale factor $k$ of the dilation is the ratio of the lengths of corresponding sides: $k = \frac{\text{Length of side } C'D'}{\text{Length of side } CD} = \frac{4}{8} = \frac{1}{2}.$

Thus, the scale factor of the dilation is $\frac{1}{2}$.

Would you like more details, or do you have any questions about this process?

Related Questions:

1. What is the center of dilation in this case?
2. How would you find the scale factor if you were given areas instead of lengths?
3. How does the orientation of the rectangle change under dilation?
4. What is the relationship between the coordinates of the points and the scale factor?
5. How can you verify that the dilation is correct using other pairs of corresponding points?

Tip: In dilation, all the points move along lines that pass through a common center point, and the distances between the points and the center scale by the same factor.