### Part A: Determining the Rotations
Given the initial triangle [tex]\( QRS \)[/tex] with vertices at [tex]\( Q(-2, -2) \)[/tex], [tex]\( R(-6, -6) \)[/tex], and [tex]\( S(-5, -1) \)[/tex], which transforms to the image triangle [tex]\( Q^{\prime}R^{\prime}S^{\prime} \)[/tex] with vertices at [tex]\( Q^{\prime}(2, -2) \)[/tex], [tex]\( R^{\prime}(6, -6) \)[/tex], [tex]\( S^{\prime}(1, -5) \)[/tex], we need to determine the possible rotations that map [tex]\( QRS \)[/tex] to [tex]\( Q^{\prime}R^{\prime}S^{\prime} \)[/tex].
After analyzing the transformations, the two possible rotations that achieve this mapping are:
- Counterclockwise rotation of [tex]\( 90^\circ \)[/tex]
- Clockwise rotation of [tex]\( 90^\circ \)[/tex]
### Part B: Verification of the Rotations
To verify the rotations, follow these steps:
1. Find the Centroids: Calculate the centroids (geometric centers) of both triangles. The centroid [tex]\(G\)[/tex] of a triangle with vertices [tex]\((x_1, y_1)\)[/tex], [tex]\((x_2, y_2)\)[/tex], and [tex]\((x_3, y_3)\)[/tex] is given by:
[tex]\[
G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
\][/tex]
2. Translate the Triangles to the Origin: Translate both triangles so that their centroids coincide with the origin [tex]\((0, 0)\)[/tex]. This helps simplify the calculation since rotations are easier to visualize and compute about the origin.
3. Calculate the Angles: For each vertex in both the original and the rotated triangles, calculate the angle each vertex makes with respect to the origin. The angle [tex]\( \theta \)[/tex] for a point [tex]\((x, y)\)[/tex] is given by:
[tex]\[
\theta = \arctan2(y, x)
\][/tex]
4. Determine Rotation Angles: Calculate the difference in angles between corresponding points in the initial and rotated triangles. The mean of these differences gives the overall angle of rotation.
5. Account for Direction: Since rotations can be either clockwise or counterclockwise, consider both possibilities:
- Counterclockwise Rotation: The angle between points is positive.
- Clockwise Rotation: The angle between points is negative, but when measured from 0 to 360 degrees, it effectively adds up to the same rotational effect.
Finally, these steps confirm that the counterclockwise rotation results in [tex]\( 90^\circ \)[/tex] and the clockwise rotation results in [tex]\( 90^\circ \)[/tex].
### Explanation
To determine the rotations, we find the centroid of each set of points and translate the points so that the centroid is at the origin. This simplifies visualizing and calculating rotations as it standardizes the reference point for both triangles. By computing the angles that each point makes with the origin in both configurations and then finding the differences between these angles, we can determine the required rotation angles.
The two possible rotations (counterclockwise and clockwise) essentially produce the same image due to the periodic nature of angular measurement, confirming the consistency and validity of the determined rotations [tex]\( 90^\circ \)[/tex] counterclockwise and [tex]\( 90^\circ \)[/tex] clockwise.
