To find the coordinates of point [tex]\(D\)[/tex] in the pre-image given the coordinates of point [tex]\(D'\)[/tex] in the image and the translation rule, follow these steps:
1. Understand the translation rule: The translation rule given is [tex]\((x, y) \to (x-4, y+15)\)[/tex]. This means that for every original point [tex]\((x, y)\)[/tex] in the pre-image, the new point in the image is obtained by subtracting 4 from the [tex]\(x\)[/tex]-coordinate and adding 15 to the [tex]\(y\)[/tex]-coordinate.
2. Express given coordinates of [tex]\(D'\)[/tex]: The coordinates of [tex]\(D'\)[/tex] in the image are [tex]\((9, -8)\)[/tex].
3. Setup the equations based on the translation rule:
Since [tex]\(D'\)[/tex] was obtained by translating [tex]\(D\)[/tex],
[tex]\[
D' = (Dx - 4, Dy + 15)
\][/tex]
Given that [tex]\(D'\)[/tex] is [tex]\((9, -8)\)[/tex], we have:
[tex]\[
Dx - 4 = 9
\][/tex]
[tex]\[
Dy + 15 = -8
\][/tex]
4. Solve for [tex]\(Dx\)[/tex]:
Add 4 to both sides of the equation [tex]\(Dx - 4 = 9\)[/tex]:
[tex]\[
Dx - 4 + 4 = 9 + 4
\][/tex]
[tex]\[
Dx = 13
\][/tex]
5. Solve for [tex]\(Dy\)[/tex]:
Subtract 15 from both sides of the equation [tex]\(Dy + 15 = -8\)[/tex]:
[tex]\[
Dy + 15 - 15 = -8 - 15
\][/tex]
[tex]\[
Dy = -23
\][/tex]
6. Conclusion:
The coordinates of point [tex]\(D\)[/tex] in the pre-image are [tex]\((13, -23)\)[/tex].
Thus, the coordinates of point [tex]\(D\)[/tex] in the pre-image are [tex]\((13, -23)\)[/tex].
