To find the coordinates of point [tex]\( D \)[/tex] (the pre-image) given the coordinates of [tex]\( D' \)[/tex] (the image) and the translation rule, we need to reverse the translation process.
The given translation rule is:
[tex]\[ (x, y) \rightarrow (x-4, y+15) \][/tex]
Given the coordinates of [tex]\( D' \)[/tex] as [tex]\((9, -8)\)[/tex], we need to apply the reverse of the translation rule to find the original coordinates [tex]\( D \)[/tex].
The translation rule tells us how to move from [tex]\( D \)[/tex] to [tex]\( D' \)[/tex]:
[tex]\[ x' = x - 4 \][/tex]
[tex]\[ y' = y + 15 \][/tex]
To find the original coordinates [tex]\( (x, y) \)[/tex] from [tex]\( (x', y') \)[/tex], we solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given the coordinates of [tex]\( D' \)[/tex]:
1. For the x-coordinate:
[tex]\[ x' = x - 4 \][/tex]
[tex]\[ 9 = x - 4 \][/tex]
[tex]\[ x = 9 + 4 \][/tex]
[tex]\[ x = 13 \][/tex]
2. For the y-coordinate:
[tex]\[ y' = y + 15 \][/tex]
[tex]\[ -8 = y + 15 \][/tex]
[tex]\[ y = -8 - 15 \][/tex]
[tex]\[ y = -23 \][/tex]
So, the coordinates of point [tex]\( D \)[/tex] are:
[tex]\[ (13, -23) \][/tex]
Hence, the correct coordinates of point [tex]\( D \)[/tex] in the pre-image are [tex]\( \boxed{(13, -23)} \)[/tex].
