To determine the vertices of triangle [tex]\( \triangle ABC \)[/tex] based on the given reflection about the line [tex]\( y = -x \)[/tex], follow these steps:
1. Understand the Reflection Rule: When a point [tex]\( (x, y) \)[/tex] is reflected over the line [tex]\( y = -x \)[/tex], its new coordinates become [tex]\( (-y, -x) \)[/tex].
2. Apply the Reflection to Each Vertex:
Given vertices of [tex]\( \triangle A'B'C' \)[/tex]:
- [tex]\( A'(-1, 1) \)[/tex]
- [tex]\( B'(-2, -1) \)[/tex]
- [tex]\( C'(-1, 0) \)[/tex]
Reflect these points about the line [tex]\( y = -x \)[/tex]:
- For [tex]\( A'(-1, 1) \)[/tex]:
[tex]\[
A = (-1, 1) \xrightarrow[\text{reflection}]{} (-1, 1)
\][/tex]
- For [tex]\( B'(-2, -1) \)[/tex]:
[tex]\[
B = (-2, -1) \xrightarrow[\text{reflection}]{} (1, 2)
\][/tex]
- For [tex]\( C'(-1, 0) \)[/tex]:
[tex]\[
C = (-1, 0) \xrightarrow[\text{reflection}]{} (0, 1)
\][/tex]
3. Identify the Correct Set of Vertices:
From the above reflections, we have the vertices of triangle [tex]\( \triangle ABC \)[/tex] as:
- [tex]\( A(-1, 1) \)[/tex]
- [tex]\( B(1, 2) \)[/tex]
- [tex]\( C(0, 1) \)[/tex]
Now, compare these vertices with the given options:
- Option A: [tex]\( A(1, -1), B(-1, -2), C(0, -1) \)[/tex]
- Option B: [tex]\( A(-1, 1), B(1, 2), C(0, 1) \)[/tex]
- Option C: [tex]\( A(-1, -1), B(-2, -1), C(-1, 0) \)[/tex]
- Option D: [tex]\( A(1, 1), B(2, -1), C(1, 0) \)[/tex]
- Option E: [tex]\( A(1, 2), B(-1, 1), C(0, 1) \)[/tex]
Clearly, Option B matches with our reflected vertices:
- [tex]\( A(-1, 1) \)[/tex]
- [tex]\( B(1, 2) \)[/tex]
- [tex]\( C(0, 1) \)[/tex]
Thus, the correct answer is Option B: [tex]\( \boxed{2} \)[/tex].
