To determine the vertices of the original triangle [tex]\(\triangle ABC\)[/tex] before the reflection about the line [tex]\(y = -x\)[/tex], we need to understand the properties of reflections.
When a point [tex]\((x', y')\)[/tex] is reflected over the line [tex]\(y = -x\)[/tex], its coordinates transform into [tex]\((y', x')\)[/tex].
Given the vertices of [tex]\(\triangle A'B'C'\)[/tex] after reflection:
- [tex]\(A'(-1, 1)\)[/tex]
- [tex]\(B'(-2, -1)\)[/tex]
- [tex]\(C'(-1, 0)\)[/tex]
We can find the original vertices by reflecting these points back over the line [tex]\(y = -x\)[/tex]:
1. For [tex]\(A'(-1, 1)\)[/tex]:
- The original vertex [tex]\(A\)[/tex] will have coordinates [tex]\((y', x') = (1, -1)\)[/tex].
- Therefore, [tex]\(A(1, -1)\)[/tex].
2. For [tex]\(B'(-2, -1)\)[/tex]:
- The original vertex [tex]\(B\)[/tex] will have coordinates [tex]\((y', x') = (-1, -2)\)[/tex].
- Therefore, [tex]\(B(-1, -2)\)[/tex].
3. For [tex]\(C'(-1, 0)\)[/tex]:
- The original vertex [tex]\(C\)[/tex] will have coordinates [tex]\((y', x') = (0, -1)\)[/tex].
- Therefore, [tex]\(C(0, -1)\)[/tex].
Thus, the vertices of [tex]\(\triangle ABC\)[/tex] are:
- [tex]\(A(1, -1)\)[/tex]
- [tex]\(B(-1, -2)\)[/tex]
- [tex]\(C(0, -1)\)[/tex]
Comparing this with the given options, the correct answer is:
A. [tex]\(A(1,-1) ; B(-1,-2), C(0,-1)\)[/tex]
