To perform a 180° rotation clockwise about the origin on a figure in the coordinate plane, you essentially need to reflect the figure across the origin. This entails flipping the figure across both the x-axis and y-axis, which results in changing the sign of both the x and y coordinates for each vertex of the triangle.
Let's perform this transformation on each vertex of triangle ABC:
1. Point A(-3, 0):
After rotating 180° clockwise about the origin:
- The x-coordinate changes sign, so -3 becomes 3.
- The y-coordinate changes sign, so 0 remains 0.
Therefore, A' is (3, 0).
2. Point B(-2, 3):
After rotating 180° clockwise about the origin:
- The x-coordinate changes sign, so -2 becomes 2.
- The y-coordinate changes sign, so 3 becomes -3.
Therefore, B' is (2, -3).
3. Point C(-1, 1):
After rotating 180° clockwise about the origin:
- The x-coordinate changes sign, so -1 becomes 1.
- The y-coordinate changes sign, so 1 becomes -1.
Therefore, C' is (1, -1).
Therefore, the coordinates of the vertices of the image after a 180° clockwise rotation about the origin are A'(3, 0), B'(2, -3), and C'(1, -1), which does not match any of the given options exactly. However, if there is a typographical error in the provided options, the closest to the correct answer would be option D, with an adjustment to the signs and order of the points.
