Answer:
To find the resultant line segment, we'll perform the translation, rotation, and scaling on point A and point B individually, and then connect the transformed points to form the new line segment.
1. **Translation by (-3, 1):**
- A' = A + (-3, 1) = (3 - 3, 2 + 1) = (0, 3)
- B' = B + (-3, 1) = (-2 - 3, -3 + 1) = (-5, -2)
2. **Rotation counterclockwise 90 degrees around (0, 3):**
- To rotate a point (x, y) counterclockwise by 90 degrees around (0, 3), the new coordinates are (-y + 3, x).
- A'' = (-2 + 3, 0 + 3) = (1, 3)
- B'' = (-(-2) + 3, -5 + 3) = (5, -2)
3. **Scaling with a factor of 2 with respect to (1, 2):**
- To scale a point (x, y) with respect to (1, 2) by a factor of 2, the new coordinates are ((2x - 1), (2y - 2)).
- A''' = (2(1) - 1, 2(3) - 2) = (1, 4)
- B''' = (2(5) - 1, 2(-2) - 2) = (9, -6)
So, the resultant line segment has endpoints A'''(1, 4) and B'''(9, -6).
