To solve the problem of finding the new coordinates of point [tex]\( P \)[/tex] after a translation, we will follow the given translation rule: [tex]\((x, y) \rightarrow (x - 5, y - 2)\)[/tex].
1. Identify the initial coordinates of point [tex]\( P \)[/tex]:
Point [tex]\( P \)[/tex] has initial coordinates [tex]\((6, -1)\)[/tex].
2. Apply the translation rule:
According to the translation rule, each coordinate of point [tex]\( P \)[/tex] needs to be adjusted as follows:
- The [tex]\( x \)[/tex]-coordinate is decreased by 5.
- The [tex]\( y \)[/tex]-coordinate is decreased by 2.
3. Calculate the new [tex]\( x \)[/tex]-coordinate:
- Initial [tex]\( x \)[/tex]-coordinate: 6
- Translation rule: [tex]\( x \rightarrow x - 5 \)[/tex]
- New [tex]\( x \)[/tex]-coordinate: [tex]\( 6 - 5 = 1 \)[/tex]
4. Calculate the new [tex]\( y \)[/tex]-coordinate:
- Initial [tex]\( y \)[/tex]-coordinate: -1
- Translation rule: [tex]\( y \rightarrow y - 2 \)[/tex]
- New [tex]\( y \)[/tex]-coordinate: [tex]\( -1 - 2 = -3 \)[/tex]
5. Write down the new coordinates of [tex]\( P' \)[/tex]:
The new coordinates of [tex]\( P' \)[/tex] after applying the translation are [tex]\((1, -3)\)[/tex].
Therefore, the coordinates of [tex]\( P' \)[/tex] are:
[tex]\[ P' (1, -3) \][/tex]
Hence, the correct answer is:
[tex]\[ P^{\prime} (1, -3) \][/tex]
