To determine the rule describing the transformation, we need to analyze how each point [tex]\( (x, y) \)[/tex] is transformed into its new position [tex]\( (x', y') \)[/tex]. Specifically, since we are looking at dilation, we need to find the dilation factor.
Given points:
[tex]\[
E(-2,-1), F(-1,1), G(2,0)
\][/tex]
Transformed points:
[tex]\[
E'(-5,-2.5), F'(-2.5,2.5), G'(5,0)
\][/tex]
### Step-by-Step Solution:
#### Step 1: Understand the Concept of Dilation
Dilation is a transformation that produces an image that is the same shape as the original, but is resized by a scale factor. The scale factor, [tex]\( k \)[/tex], is the ratio of a coordinate of the image to the corresponding coordinate of the pre-image.
#### Step 2: Calculate the Scale Factor for Each Point
1. Calculate the scale factor using point [tex]\( E \)[/tex] and [tex]\( E' \)[/tex]:
[tex]\[
k_E = \frac{E_x'}{E_x} = \frac{-5}{-2} = 2.5 \quad \text{and} \quad k_E = \frac{E_y'}{E_y} = \frac{-2.5}{-1} = 2.5
\][/tex]
Both [tex]\( k_E \)[/tex] values are equal to 2.5.
2. Calculate the scale factor using point [tex]\( F \)[/tex] and [tex]\( F' \)[/tex]:
[tex]\[
k_F = \frac{F_x'}{F_x} = \frac{-2.5}{-1} = 2.5 \quad \text{and} \quad k_F = \frac{F_y'}{F_y} = \frac{2.5}{1} = 2.5
\][/tex]
Both [tex]\( k_F \)[/tex] values are equal to 2.5.
3. Calculate the scale factor using point [tex]\( G \)[/tex] and [tex]\( G' \)[/tex]:
[tex]\[
k_G = \frac{G_x'}{G_x} = \frac{5}{2} = 2.5 \quad \text{and} \quad k_G = \frac{G_y'}{G_y} = \frac{0}{0}
\][/tex]
Since [tex]\( G_y = 0 \)[/tex] and [tex]\( G_y' = 0 \)[/tex], this part is not useful for calculating the scale factor, but [tex]\( k_G \)[/tex] for [tex]\( x \)[/tex]-coordinates confirms that it is 2.5.
#### Step 3: Conclusion
Since all computed dilation factors for the respective points are consistent, we conclude that the dilation factor is [tex]\( 2.5 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\text{Dilation of } 2.5
\][/tex]
