To determine the coordinates of vertex [tex]\( V \)[/tex] of the pre-image given the dilation rule [tex]\( D_{O, 133}(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right) \)[/tex], we need to reverse this dilation rule. This means we need to find the original coordinates [tex]\((x, y)\)[/tex] that, when transformed by the dilation rule, result in the coordinates of vertex [tex]\( V \)[/tex] of the image.
Let's operate on each of the provided options to see which one corresponds to the given dilation rule.
### Step-by-Step Solution:
1. Option 1: [tex]\((0,0)\)[/tex]
- Applying the reverse dilation:
[tex]\[
(0, 0) \rightarrow (0 \cdot 3, 0 \cdot 3) = (0, 0)
\][/tex]
- The pre-image coordinates for the image point [tex]\((0,0)\)[/tex] are [tex]\((0,0)\)[/tex].
2. Option 2: [tex]\(\left(0, \frac{1}{3}\right)\)[/tex]
- Applying the reverse dilation:
[tex]\[
\left(0, \frac{1}{3}\right) \rightarrow \left(0 \cdot 3, \frac{1}{3} \cdot 3\right) = (0, 1)
\][/tex]
- The pre-image coordinates for the image point [tex]\(\left(0, \frac{1}{3}\right)\)[/tex] are [tex]\((0, 1)\)[/tex].
3. Option 3: [tex]\((0,1)\)[/tex]
- Applying the reverse dilation:
[tex]\[
(0, 1) \rightarrow (0 \cdot 3, 1 \cdot 3) = (0, 3)
\][/tex]
- The pre-image coordinates for the image point [tex]\((0, 1)\)[/tex] are [tex]\((0, 3)\)[/tex].
4. Option 4: [tex]\((0,3)\)[/tex]
- Applying the reverse dilation:
[tex]\[
(0, 3) \rightarrow (0 \cdot 3, 3 \cdot 3) = (0, 9)
\][/tex]
- The pre-image coordinates for the image point [tex]\((0, 3)\)[/tex] are [tex]\((0, 9)\)[/tex].
After examining all the options, the coordinates of vertex [tex]\( V \)[/tex] in the pre-image must correspond to the coordinates [tex]\((0, 3)\)[/tex] in the original image under the given dilation rule. Therefore, option 3, which is [tex]\((0, 1)\)[/tex], is the one that translates to [tex]\((0, 3)\)[/tex] in the pre-image when the dilation rule is reversed.
Thus, the coordinates of vertex [tex]\( V \)[/tex] of the pre-image are:
[tex]\[
\boxed{(0, 3)}
\][/tex]
