Alright, let's solve the problem step-by-step. We know that square [tex]\(ABCD\)[/tex] was translated following the rule [tex]\((x, y) \rightarrow (x-4, y+15)\)[/tex], and we are given the coordinates of the translated point [tex]\(D'\)[/tex] as [tex]\((9, -8)\)[/tex]. Our aim is to find the coordinates of the original point [tex]\(D\)[/tex] before the translation, denoted simply as [tex]\(D\)[/tex].
Given:
- Translation rule: [tex]\((x, y) \rightarrow (x-4, y+15)\)[/tex]
- Coordinates of [tex]\(D'\)[/tex]: [tex]\((9, -8)\)[/tex]
To find the coordinates of the original point [tex]\(D\)[/tex], we need to reverse the translation process.
1. Let's denote the coordinates of the original point [tex]\(D\)[/tex] as [tex]\((x, y)\)[/tex].
2. According to the translation rule, the x-coordinate of [tex]\(D'\)[/tex] is obtained by subtracting 4 from the x-coordinate of [tex]\(D\)[/tex]:
[tex]\[ x - 4 \rightarrow 9 \][/tex]
So,
[tex]\[ x - 4 = 9 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = 9 + 4 \][/tex]
[tex]\[ x = 13 \][/tex]
3. Similarly, the y-coordinate of [tex]\(D'\)[/tex] is obtained by adding 15 to the y-coordinate of [tex]\(D\)[/tex]:
[tex]\[ y + 15 \rightarrow -8 \][/tex]
So,
[tex]\[ y + 15 = -8 \][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[ y = -8 - 15 \][/tex]
[tex]\[ y = -23 \][/tex]
Therefore, the coordinates of point [tex]\(D\)[/tex] in the pre-image are [tex]\((13, -23)\)[/tex].
Hence, the correct answer is:
[tex]\[
(13, -23)
\][/tex]
