To solve this problem, let's break it down into two main transformations: a 180-degree clockwise rotation about the origin, followed by a reflection across the line [tex]\( y = -x \)[/tex].
### Step 1: 180-Degree Clockwise Rotation
The coordinates of the vertices [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex] of [tex]\( \triangle ABC \)[/tex] are given as:
- [tex]\( A(-3, 0) \)[/tex]
- [tex]\( B(-2, 3) \)[/tex]
- [tex]\( C(-1, 1) \)[/tex]
For a 180-degree clockwise rotation around the origin, we use the formula:
[tex]\[ (x, y) \rightarrow (-x, -y) \][/tex]
Applying this to each vertex:
- For [tex]\( A(-3, 0) \)[/tex]:
[tex]\[ A' = (3, 0) \][/tex]
- For [tex]\( B(-2, 3) \)[/tex]:
[tex]\[ B' = (2, -3) \][/tex]
- For [tex]\( C(-1, 1) \)[/tex]:
[tex]\[ C' = (1, -1) \][/tex]
Thus, the coordinates after the 180-degree rotation are:
- [tex]\( A' = (3, 0) \)[/tex]
- [tex]\( B' = (2, -3) \)[/tex]
- [tex]\( C' = (1, -1) \)[/tex]
### Step 2: Reflection Across the Line [tex]\( y = -x \)[/tex]
For reflection over the line [tex]\( y = -x \)[/tex], we use the formula:
[tex]\[ (x, y) \rightarrow (-y, -x) \][/tex]
Applying this to each vertex:
- For [tex]\( A'(3, 0) \)[/tex]:
[tex]\[ A'' = (0, 3) \][/tex]
- For [tex]\( B'(2, -3) \)[/tex]:
[tex]\[ B'' = (-3, -2) \][/tex]
- For [tex]\( C'(1, -1) \)[/tex]:
[tex]\[ C'' = (-1, -1) \][/tex]
So, the coordinates after reflection are:
- [tex]\( A'' = (0, 3) \)[/tex]
- [tex]\( B'' = (-3, -2) \)[/tex]
- [tex]\( C'' = (-1, -1) \)[/tex]
### Conclusion
The final coordinates of the vertices after both transformations are:
- [tex]\( A'' = (0, 3) \)[/tex]
- [tex]\( B'' = (-3, -2) \)[/tex]
- [tex]\( C'' = (-1, -1) \)[/tex]
From the options given:
- Option D is the correct answer:
[tex]\[
A'' = (0, 3), \, B'' = (-3,-2), \, C'' = (-1, -1)
\][/tex]
