Part A: The two different rotations that would create the image are:
1. A 180-degree rotation around the origin (0,0).
2. A 180-degree rotation around the point (-4, -4).
Part B: We know our answer is correct because both rotations preserve the distances between the original vertices, the angle measures, the orientation of the triangle, and are consistent with the given image coordinates.
Part A: To find the rotations that would create the image, we need to analyze the changes in coordinates from the original triangle QRS to its image Q′R′S′.
1. **Rotation around the origin (0,0):**
- Q(−2, −2) becomes Q′(2, −2)
- R(−6, −6) becomes R′(6, −6)
- S(−5, −1) becomes S′(1, −5)
2. **Rotation around the point (−4, −4):**
- Q(−2, −2) becomes Q′(2, −2)
- R(−6, −6) becomes R′(6, −6)
- S(−5, −1) becomes S′(1, −5)
Both rotations yield the same image vertices, so either of these rotations could be responsible for the transformation.
Part B: We can verify the correctness of our answer by checking the properties of rotations:
1. **Preservation of Distance:** In both cases, the distance between each pair of corresponding points remains the same. For example, the distance between Q and R is the same as the distance between Q′ and R′.
2. **Preservation of Angle Measures:** The angles between the sides are preserved in both rotations. For example, the angle between QR and RS is the same as the angle between Q′R′ and R′S′.
3. **Preservation of Orientation:** The order of the vertices (clockwise or counterclockwise) is preserved in both rotations. In this case, both rotations maintain the counterclockwise orientation of the vertices.
4. **Consistency with Given Coordinates:** Both rotations correctly transform the given coordinates of Q, R, and S to the coordinates of Q′, R′, and S′.
Therefore, either of the rotations around the origin or around the point (-4, -4) would create the given image Q′R′S′.