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Question

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 9 to 9 in increments of 1. Three rectangles are formed by lines joining four plotted points each. The coordinates of the plotted points for first rectangle joined by solid lines are left parenthesis negative 5 comma negative 4 right parenthesis, left parenthesis negative 1 comma negative 4 right parenthesis, left parenthesis negative 1 comma negative 6 right parenthesis, and left parenthesis negative 5 comma negative 6 right parenthesis. The coordinates for the second rectangle joined by lines made of dashes and dots are left parenthesis 3 comma 4 right parenthesis, left parenthesis 5 comma 4 right parenthesis, left parenthesis 5 comma 8 right parenthesis, and left parenthesis 3 comma 8 right parenthesis. The coordinates of the plotted points for the third rectangle joined by dotted lines are left parenthesis negative 4 comma 1 right parenthesis, left parenthesis negative 6 comma 1 right parenthesis, left parenthesis negative 6 comma 5 right parenthesis, and left parenthesis negative 4 comma 5 right parenthesis.
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. What is the next step?

Answer 1

To determine the congruence of the rectangles, after rotating the rectangle in Quadrant III (the first rectangle) 90 degrees clockwise about the origin, the next step would be to translate the rotated rectangle to align it with the coordinates of the rectangle in Quadrant I.

This translation process would involve moving the rotated rectangle along the coordinate plane until its vertices match the corresponding vertices of the rectangle in Quadrant I. Specifically, you would find the required horizontal and vertical shifts needed to position the rotated rectangle over the rectangle defined by the coordinates in Quadrant I.

Once the rectangles are aligned, you can then compare their dimensions (length and width) to confirm that they are congruent.