To determine the location of the point [tex]\( Z^ \)[/tex] after performing the required transformations on the point [tex]\( Z \)[/tex], we follow a step-by-step process of translations and reflections:
1. Identify the original coordinates of point [tex]\( Z \)[/tex]:
The given coordinates of point [tex]\( Z \)[/tex] are [tex]\((8, 2)\)[/tex].
2. Translate the point [tex]\( Z \)[/tex] using the rule [tex]\((x, y) \rightarrow (x+2, y+3)\)[/tex]:
- New [tex]\( x \)[/tex]-coordinate:
[tex]\[ x + 2 = 8 + 2 = 10 \][/tex]
- New [tex]\( y \)[/tex]-coordinate:
[tex]\[ y + 3 = 2 + 3 = 5 \][/tex]
After the translation, the new coordinates of the point [tex]\( Z \)[/tex] are [tex]\((10, 5)\)[/tex].
3. Reflect the translated point across the [tex]\( x \)[/tex]-axis:
Reflecting a point across the [tex]\( x \)[/tex]-axis changes the sign of the [tex]\( y \)[/tex]-coordinate, while the [tex]\( x \)[/tex]-coordinate remains the same.
- [tex]\( x \)[/tex]-coordinate remains:
[tex]\[ x = 10 \][/tex]
- [tex]\( y \)[/tex]-coordinate changes sign:
[tex]\[ y = -5 \][/tex]
After the reflection, the coordinates of the point [tex]\( Z \)[/tex] are [tex]\((10, -5)\)[/tex].
Therefore, the coordinates of [tex]\( Z^ \)[/tex] are [tex]\((10, -5)\)[/tex].
Given the options:
- [tex]\((2, -8)\)[/tex]
- [tex]\((6, -2)\)[/tex]
- [tex]\((8, -2)\)[/tex]
- [tex]\((12, -6)\)[/tex]
None of them match our calculated point [tex]\((10, -5)\)[/tex]. There might have been a typo or miscalculation in the interpretation of the options, but based on our solution, [tex]\( Z^* \)[/tex] is at [tex]\((10, -5)\)[/tex].
