The series of transformations that have occurred to move the triangle in quadrant 4 to the triangle in quadrant 2 are:
D) rotation, then translation
Explanation:
Step 1: Rotation
The first transformation that occurred is a rotation. The triangle is rotated 180 degrees about the origin (0,0). This moves the original triangle in quadrant 4 to quadrant 2. The new coordinates after rotation are:
G' (-5, -5)
E' (-1, -3)
H' (-5, -3)
Step 2: Translation
The second transformation is a translation. The triangle is moved 8 units in the positive x-direction and 2 units in the positive y-direction. This gives us the final new position of the triangle in quadrant 2:
G (3, 3)
E (7, 1)
H (3, 1)
Therefore, the correct answer is D) rotation, then translation.