Sure, let's tackle this step-by-step:
1. Understand the dilation rule: The given rule is [tex]\( D_{O, \frac{1}{3}}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right) \)[/tex]. This means that to get the image coordinates from the pre-image coordinates, you multiply each coordinate by [tex]\(\frac{1}{3}\)[/tex].
2. Formulate the reverse process: To find the pre-image from the image, we need to reverse this operation. Instead of multiplying by [tex]\(\frac{1}{3}\)[/tex], we multiply by the reciprocal, which is 3.
3. Identify the image coordinates: Assume the coordinates of [tex]\( V' \)[/tex] in the image are given as [tex]\( (0, 3) \)[/tex].
4. Apply the reverse dilation:
- For the [tex]\( x \)[/tex]-coordinate of [tex]\( V' \)[/tex], you reverse the dilation operation:
[tex]\[
x = 3 \times 0 = 0
\][/tex]
- For the [tex]\( y \)[/tex]-coordinate of [tex]\( V' \)[/tex], similarly:
[tex]\[
y = 3 \times 3 = 9
\][/tex]
5. Determine the pre-image coordinates: After applying the reverse dilation operation, the coordinates of the pre-image vertex [tex]\( V \)[/tex] are [tex]\( (0, 9) \)[/tex].
6. Check the given options: The given options are:
- [tex]\( (0,0) \)[/tex]
- [tex]\( \left(0, \frac{1}{3}\right) \)[/tex]
- [tex]\( (0,1) \)[/tex]
- [tex]\( (0,3) \)[/tex]
None of the provided options match the determined pre-image coordinates of [tex]\( V \)[/tex], which are [tex]\( (0, 9) \)[/tex].
Given this comprehensive step-by-step analysis, there seems to be an issue, as none of the provided options are correct based on the given rules and analysis. However, following the given correct solution:
The coordinates of vertex [tex]\( V \)[/tex] of the pre-image are [tex]\( (0, 9) \)[/tex].
This is consistently deduced from reversing the dilation of the given coordinates [tex]\( (0, 3) \)[/tex].
