Let's solve the problem step-by-step:
1. Initial Coordinates:
- Given points are [tex]\( A(3, 2) \)[/tex] and [tex]\( B(2, 3) \)[/tex].
2. Rotation by 180° about the Origin:
- When a point [tex]\((x, y)\)[/tex] is rotated 180° about the origin, its new coordinates become [tex]\((-x, -y)\)[/tex].
- For point [tex]\( A(3, 2) \)[/tex]:
- The new coordinates after rotation are [tex]\((-3, -2)\)[/tex].
- For point [tex]\( B(2, 3) \)[/tex]:
- The new coordinates after rotation are [tex]\((-2, -3)\)[/tex].
After the 180° rotation, the points are:
- [tex]\(A'(-3, -2)\)[/tex]
- [tex]\(B'(-2, -3)\)[/tex]
3. Reflection across the y-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the y-axis, the x-coordinate is negated while the y-coordinate remains unchanged.
- For point [tex]\( A'(-3, -2) \)[/tex]:
- The new coordinates after reflection are [tex]\((3, -2)\)[/tex].
- For point [tex]\( B'(-2, -3) \)[/tex]:
- The new coordinates after reflection are [tex]\((2, -3)\)[/tex].
After reflection across the y-axis, the final points are:
- [tex]\(A''(3, -2)\)[/tex]
- [tex]\(B''(2, -3)\)[/tex]
Therefore, after rotating the line segment [tex]\( AB \)[/tex] about the origin through 180° and then reflecting it across the y-axis, the new coordinates of points A and B are [tex]\((3, -2)\)[/tex] and [tex]\((2, -3)\)[/tex] respectively.
