After Darnell rotates the rectangle in Quadrant III (the first rectangle) 90 degrees clockwise into Quadrant II, his next step would be to translate the rotated rectangle horizontally and vertically to match the position of the rectangle in Quadrant I (the second rectangle).
To determine the translation, we need to find the appropriate horizontal and vertical movement to align one rectangle with the other. The coordinates of the rectangle in Quadrant I are:
- (2, 3)
- (7, 3)
- (7, 5)
- (2, 5)
For the rectangle from Quadrant III (originally defined by its coordinates):
- (-1, -2)
- (-3, -2)
- (-3, -7)
- (-1, -7)
After a 90-degree clockwise rotation, the transformed coordinates of this rectangle will correspond to points in Quadrant II. The translation would then involve calculating how far to move each point to align with the points in Quadrant I.
To translate the rotated rectangle so that its new position aligns with the rectangle in Quadrant I, the translation vector can be determined by finding the difference between the corresponding points. For example, moving the bottom-left corner from its new location in Quadrant II to the bottom-left corner in Quadrant I would give a translation vector.
Thus, the next step is to translate the points by that determined vector, which you would represent as:
translate (x, y)
Where (x, y) is the specific translation coordinates found after determining the correct offset needed to align the two rectangles after rotation.