To determine the vertices of the pre-image rectangle [tex]\(ABCD\)[/tex], we need to reverse the translation that was applied to the image rectangle [tex]\(A'B'C'D'\)[/tex]. The image rectangle was translated according to the rule [tex]\(T_{-4, 3}(x, y) = (x - 4, y + 3)\)[/tex].
Here are the steps to find the original vertices:
1. Identify the translation rule:
- The rule [tex]\(T_{-4, 3}(x, y)\)[/tex] indicates that each [tex]\(x\)[/tex]-coordinate of the image rectangle was obtained by subtracting 4 from the [tex]\(x\)[/tex]-coordinate of the original rectangle.
- Each [tex]\(y\)[/tex]-coordinate of the image rectangle was obtained by adding 3 to the [tex]\(y\)[/tex]-coordinate of the original rectangle.
2. Given translated points (vertices of [tex]\(A'B'C'D'\)[/tex]):
- [tex]\((-1, -2)\)[/tex]
- [tex]\( (7, 1)\)[/tex]
- [tex]\((-1, 7)\)[/tex]
- [tex]\((-1, 1)\)[/tex]
- [tex]\((7, -2)\)[/tex]
3. Reverse the translation to find the original vertices:
- For the point [tex]\((-1, -2)\)[/tex]:
[tex]\[
x = -1 + 4 = 3 \quad \text{and} \quad y = -2 - 3 = -5
\][/tex]
So, the original point is [tex]\((3, -5)\)[/tex].
- For the point [tex]\( (7, 1)\)[/tex]:
[tex]\[
x = 7 + 4 = 11 \quad \text{and} \quad y = 1 - 3 = -2
\][/tex]
So, the original point is [tex]\((11, -2)\)[/tex].
- For the point [tex]\((-1, 7)\)[/tex]:
[tex]\[
x = -1 + 4 = 3 \quad \text{and} \quad y = 7 - 3 = 4
\][/tex]
So, the original point is [tex]\((3, 4)\)[/tex].
- For the point [tex]\((-1, 1)\)[/tex]:
[tex]\[
x = -1 + 4 = 3 \quad \text{and} \quad y = 1 - 3 = -2
\][/tex]
So, the original point is [tex]\((3, -2)\)[/tex].
- For the point [tex]\((7, -2)\)[/tex]:
[tex]\[
x = 7 + 4 = 11 \quad \text{and} \quad y = -2 - 3 = -5
\][/tex]
So, the original point is [tex]\((11, -5)\)[/tex].
Therefore, the vertices of the pre-image rectangle [tex]\(ABCD\)[/tex] are:
[tex]\[
\boxed{(3, -5), (11, -2), (3, 4), (3, -2), (11, -5)}
\][/tex]
Among these, we select the unique four vertices:
[tex]\[
(3, -5), (11, -2), (3, 4), (11, -5)
\][/tex]
