To solve this problem, we need to determine the coordinates of the image of the triangle [tex]$\triangle ABC$[/tex] after performing two transformations: a [tex]$180^{\circ}$[/tex] clockwise rotation about the origin and a reflection across the line [tex]$y=-x$[/tex]. Let's go through each step in detail:
### Step 1: Rotating [tex]$180^{\circ}$[/tex] clockwise about the origin
When a point [tex]$(x, y)$[/tex] is rotated [tex]$180^{\circ}$[/tex] clockwise about the origin, the new coordinates become [tex]$(-x, -y)$[/tex].
The original coordinates of the vertices are:
- [tex]\( A(-3, 0) \)[/tex]
- [tex]\( B(-2, 3) \)[/tex]
- [tex]\( C(-1, 1) \)[/tex]
Applying the [tex]$180^{\circ}$[/tex] clockwise rotation:
- For vertex [tex]\( A \)[/tex]:
[tex]\[
A' = (-(-3), -(0)) = (3, 0)
\][/tex]
- For vertex [tex]\( B \)[/tex]:
[tex]\[
B' = (-(-2), -(3)) = (2, -3)
\][/tex]
- For vertex [tex]\( C \)[/tex]:
[tex]\[
C' = (-(-1), -(1)) = (1, -1)
\][/tex]
After the [tex]$180^{\circ}$[/tex] rotation, the coordinates of the vertices are:
- [tex]\( A'(3, 0) \)[/tex]
- [tex]\( B'(2, -3) \)[/tex]
- [tex]\( C'(1, -1) \)[/tex]
### Step 2: Reflecting across the line [tex]$y = -x$[/tex]
When a point [tex]$(x, y)$[/tex] is reflected across the line [tex]$y = -x$[/tex], the new coordinates become [tex]$(y, x)$[/tex].
Using the new coordinates from the rotation step:
- For vertex [tex]\( A' \)[/tex]:
[tex]\[
A'' = (0, 3)
\][/tex]
- For vertex [tex]\( B' \)[/tex]:
[tex]\[
B'' = (-3, 2)
\][/tex]
- For vertex [tex]\( C' \)[/tex]:
[tex]\[
C'' = (-1, 1)
\][/tex]
After reflecting across the line [tex]$y = -x$[/tex], the coordinates of the vertices are:
- [tex]\( A''(0, 3) \)[/tex]
- [tex]\( B''(-3, 2) \)[/tex]
- [tex]\( C''(-1, 1) \)[/tex]
Thus, the correct answer is:
### A. [tex]\( A''(0, 3), B''(-3, 2), C''(-1, 1) \)[/tex]
