To solve this problem, we need to apply two transformations to the given triangle vertices [tex]\( A(-3,0), B(-2,3), C(-1,1) \)[/tex]:
1. Rotate [tex]\( \triangle ABC \)[/tex] by [tex]\( 180^\circ \)[/tex] clockwise about the origin.
2. Reflect the resulting image across the line [tex]\( y = -x \)[/tex].
Step 1: Rotate by [tex]\( 180^\circ \)[/tex] clockwise
When a point [tex]\((x, y)\)[/tex] is rotated by [tex]\( 180^\circ \)[/tex] clockwise about the origin, the new coordinates [tex]\((x', y')\)[/tex] will be [tex]\((-x, -y)\)[/tex].
Thus, we have:
- [tex]\( A(-3, 0) \)[/tex] rotated by [tex]\( 180^\circ \)[/tex] becomes [tex]\( A' (3, 0) \)[/tex].
- [tex]\( B(-2, 3) \)[/tex] rotated by [tex]\( 180^\circ \)[/tex] becomes [tex]\( B' (2, -3) \)[/tex].
- [tex]\( C(-1, 1) \)[/tex] rotated by [tex]\( 180^\circ \)[/tex] becomes [tex]\( C' (1, -1) \)[/tex].
Step 2: Reflect across the line [tex]\( y = -x \)[/tex]
To reflect a point [tex]\((x, y)\)[/tex] across the line [tex]\( y = -x \)[/tex], the new coordinates [tex]\((x', y')\)[/tex] will be [tex]\((-y, -x)\)[/tex].
Applying this reflection:
- [tex]\( A' (3, 0) \)[/tex] reflected across [tex]\( y = -x \)[/tex] becomes [tex]\( A'' (0, -3) \)[/tex].
- [tex]\( B' (2, -3) \)[/tex] reflected across [tex]\( y = -x \)[/tex] becomes [tex]\( B'' (3, -2) \)[/tex].
- [tex]\( C' (1, -1) \)[/tex] reflected across [tex]\( y = -x \)[/tex] becomes [tex]\( C'' (1, -1) \)[/tex].
Thus, the coordinates of the vertices after both transformations are:
- [tex]\( A'' (0, -3) \)[/tex]
- [tex]\( B'' (3, -2) \)[/tex]
- [tex]\( C'' (1, -1) \)[/tex]
Therefore, the correct answer is:
R. [tex]\(A^{\prime}(0,-3), B(3,-2), C^{\prime}(1,-1)\)[/tex]
