To solve for the coordinates of vertex \(V\) of the pre-image given the dilation rule \(D_{O, \frac{1}{3}} (x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)\), we'll need to determine which of the given options, when dilated, results in the corresponding coordinates provided.
First, recall the dilation rule:
[tex]\[ D_{O, \frac{1}{3}} (x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right) \][/tex]
We'll analyze each given pre-image coordinate option and apply the dilation rule to see which one matches the destination coordinates.
Step-by-Step Process:
1. Option 1: \((0, 0)\)
[tex]\[ D_{O, \frac{1}{3}} (0, 0) \rightarrow \left(\frac{1}{3} \cdot 0, \frac{1}{3} \cdot 0\right) = (0, 0) \][/tex]
2. Option 2: \(\left(0, \frac{1}{3}\right)\)
[tex]\[ D_{O, \frac{1}{3}} \left(0, \frac{1}{3}\right) \rightarrow \left(\frac{1}{3} \cdot 0, \frac{1}{3} \cdot \frac{1}{3}\right) = \left(0, \frac{1}{9}\right) \][/tex]
3. Option 3: \((0, 1)\)
[tex]\[ D_{O, \frac{1}{3}} (0, 1) \rightarrow \left(\frac{1}{3} \cdot 0, \frac{1}{3} \cdot 1\right) = \left(0, \frac{1}{3}\right) \][/tex]
4. Option 4: \((0, 3)\)
[tex]\[ D_{O, \frac{1}{3}} (0, 3) \rightarrow \left(\frac{1}{3} \cdot 0, \frac{1}{3} \cdot 3\right) = (0, 1) \][/tex]
From the results, we notice that dilating \((0,0)\) by the factor of \(\frac{1}{3}\) gives the image coordinates \((0,0)\), dilating \(\left(0, \frac{1}{3}\right)\) gives \(\left(0, \frac{1}{9}\)\), dilating \((0,1)\) gives \((0, \frac{1}{3})\), and dilating \((0,3)\) gives \((0,1)\).
The correct coordinate of vertex \(V\) of the pre-image that matches with the destination coordinates is:
[tex]\[(0,0)\][/tex]
Thus, the coordinates of vertex [tex]\(V\)[/tex] of the pre-image are [tex]\(\boxed{(0,0)}\)[/tex].
