Let's follow the transformations step-by-step to find the coordinates of Point [tex]\( A^{\prime\prime} \)[/tex] for triangle [tex]\( ABC \)[/tex]:
1. Translation:
- The initial coordinates of point [tex]\( A \)[/tex] are [tex]\( (1, 1) \)[/tex].
- According to the translation rule [tex]\((x + 3, y - 2)\)[/tex]:
- The x-coordinate will be [tex]\( 1 + 3 = 4 \)[/tex].
- The y-coordinate will be [tex]\( 1 - 2 = -1 \)[/tex].
- Thus, after the translation, the coordinates of point [tex]\( A' \)[/tex] are [tex]\( (4, -1) \)[/tex].
2. Dilation:
- Now, point [tex]\( A' \)[/tex] with coordinates [tex]\( (4, -1) \)[/tex] needs to be dilated by a scale factor of 3 about the origin.
- For dilation about the origin, the new coordinates are obtained by multiplying each coordinate by the scale factor.
- The x-coordinate will be [tex]\( 4 \times 3 = 12 \)[/tex].
- The y-coordinate will be [tex]\( -1 \times 3 = -3 \)[/tex].
- Hence, after dilation, the coordinates of point [tex]\( A^{\prime\prime} \)[/tex] are [tex]\( (12, -3) \)[/tex].
Therefore, the coordinates of Point [tex]\( A^{\prime\prime} \)[/tex] are [tex]\( (12, -3) \)[/tex]. Thus, the correct answer is:
[tex]\[ A^{\prime\prime}(12, -3) \][/tex]
