To determine the scale factor used in the dilation process, we need to compare the coordinates of the pre-image point [tex]\( N (6, -3) \)[/tex] and the image point [tex]\( N' (2, -1) \)[/tex]. The scale factor, often represented by [tex]\( k \)[/tex], is a ratio that maps the pre-image coordinates to the image coordinates.
Let's analyze each coordinate separately.
### For the [tex]\( x \)[/tex]-coordinate:
Given:
- Pre-image [tex]\( x \)[/tex]-coordinate [tex]\( x = 6 \)[/tex]
- Image [tex]\( x \)[/tex]-coordinate [tex]\( x' = 2 \)[/tex]
The scale factor for the [tex]\( x \)[/tex]-coordinate, [tex]\( k_x \)[/tex], is given by:
[tex]\[ k_x = \frac{x'}{x} = \frac{2}{6} = \frac{1}{3} \][/tex]
### For the [tex]\( y \)[/tex]-coordinate:
Given:
- Pre-image [tex]\( y \)[/tex]-coordinate [tex]\( y = -3 \)[/tex]
- Image [tex]\( y \)[/tex]-coordinate [tex]\( y' = -1 \)[/tex]
The scale factor for the [tex]\( y \)[/tex]-coordinate, [tex]\( k_y \)[/tex], is given by:
[tex]\[ k_y = \frac{y'}{y} = \frac{-1}{-3} = \frac{1}{3} \][/tex]
### Consistency of Scale Factors:
We observe that both the [tex]\( x \)[/tex]-coordinate and the [tex]\( y \)[/tex]-coordinate yield the same scale factor:
[tex]\[ k_x = \frac{1}{3} \quad \text{and} \quad k_y = \frac{1}{3} \][/tex]
Since the scale factors for both coordinates are consistent, we can conclude that the overall scale factor [tex]\( k \)[/tex] for the dilation process is:
[tex]\[ k = \frac{1}{3} \][/tex]
Therefore, the scale factor used is [tex]\( \boxed{\frac{1}{3}} \)[/tex].
