To solve this problem, let's carefully follow the translation rule and work backward to determine the vertices of the pre-image rectangle [tex]\(ABCD\)[/tex].
Step-by-Step Solution:
1. Understand the Translation Rule:
The translation rule given is [tex]\( T_{-4,3}(x, y) \)[/tex]. This translates a point [tex]\((x, y)\)[/tex] to a new point [tex]\((x', y')\)[/tex] where:
[tex]\[
x' = x - 4
\][/tex]
[tex]\[
y' = y + 3
\][/tex]
2. Given Points as Image Vertices:
We have several possible vertices given for rectangle [tex]\(A'B'C'D'\)[/tex]. They include:
- [tex]\((-1, -2)\)[/tex]
- [tex]\((7, 1)\)[/tex]
- [tex]\((-1, 7)\)[/tex]
- [tex]\((-1, 1)\)[/tex]
- [tex]\((7, -2)\)[/tex]
3. Determine Pre-Image Coordinates:
To find the corresponding vertices of rectangle [tex]\(ABCD\)[/tex] before translation, we need to reverse the translation. For each point [tex]\((x', y')\)[/tex], we have to compute [tex]\((x, y)\)[/tex] where:
[tex]\[
x = x' + 4
\][/tex]
[tex]\[
y = y' - 3
\][/tex]
Let's compute the pre-image points for each given point:
- For [tex]\((-1, -2)\)[/tex]:
[tex]\[
x = -1 + 4 = 3
\][/tex]
[tex]\[
y = -2 - 3 = -5
\][/tex]
- For [tex]\((7, 1)\)[/tex]:
[tex]\[
x = 7 + 4 = 11
\][/tex]
[tex]\[
y = 1 - 3 = -2
\][/tex]
- For [tex]\((-1, 7)\)[/tex]:
[tex]\[
x = -1 + 4 = 3
\][/tex]
[tex]\[
y = 7 - 3 = 4
\][/tex]
- For [tex]\((-1, 1)\)[/tex]:
[tex]\[
x = -1 + 4 = 3
\][/tex]
[tex]\[
y = 1 - 3 = -2
\][/tex]
- For [tex]\((7, -2)\)[/tex]:
[tex]\[
x = 7 + 4 = 11
\][/tex]
[tex]\[
y = -2 - 3 = -5
\][/tex]
4. Compile Results:
The vertices of the pre-image, rectangle [tex]\(ABCD\)[/tex], are:
- [tex]\((3, -5)\)[/tex]
- [tex]\((11, -2)\)[/tex]
- [tex]\((3, 4)\)[/tex]
- [tex]\((3, -2)\)[/tex]
- [tex]\((11, -5)\)[/tex]
Given these calculations, the vertices of the original rectangle [tex]\(ABCD\)[/tex] are:
- [tex]\((3, -5)\)[/tex]
- [tex]\((11, -2)\)[/tex]
- [tex]\((3, 4)\)[/tex]
- [tex]\((3, -2)\)[/tex]
Hence, the points that match the vertices of the pre-image of the given points are:
- [tex]\((3, -5)\)[/tex]
- [tex]\((11, -2)\)[/tex]
- [tex]\((3, 4)\)[/tex]
- [tex]\((3, -2)\)[/tex]
Concluding, the vertices of the pre-image rectangle [tex]\(ABCD\)[/tex] that correspond to the given image vertices are [tex]\(\boxed{(3, -5)}, \boxed{(11, -2)}, \boxed{(3, 4)}, \boxed{(3, -2)}\)[/tex].
