The first step to determine that the rectangle in Quadrant III is congruent to the rectangle in Quadrant I was to rotate the first rectangle 90 degrees clockwise about the origin. The next step is translating it, but what coordinates do we translate it by? (The current coordinates are -1,-4 -5,-4 -1,-6 and -5,-6.)
To translate the rectangle in Quadrant III to match the rectangle in Quadrant I, we need to shift it by the same amount horizontally and vertically.
Given that the coordinates of the rectangle in Quadrant III are (-1, -4), (-5, -4), (-1, -6), and (-5, -6), we need to translate it so that its corresponding vertices in Quadrant I will be (1, 4), (5, 4), (1, 6), and (5, 6).
To determine the translation vector, we can calculate the difference between the coordinates of the two corresponding vertices in the x and y directions.
For the x-direction:
Translation in x = 1 - (-1) = 2
For the y-direction:
Translation in y = 4 - (-4) = 8
Therefore, we need to translate the rectangle in Quadrant III by (2, 8) to match the rectangle in Quadrant I.