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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

The next step is to translate the rotated rectangle so that its corresponding vertices align with those of the rectangle in Quadrant I. After rotating the Rectangle in Quadrant III (which has vertices at (-5, -4), (-1, -4), (-1, -6), and (-5, -6)) 90 degrees clockwise, the new vertices will be positioned in a different location.

For example, rotating the point (-5, -4) 90 degrees clockwise about the origin results in the point (-4, 5). You would follow this process for all points of the rectangle. After obtaining the coordinates of the rotated rectangle, you then translate or shift it by moving it to the appropriate position in the coordinate plane so that it matches the coordinates of the rectangle in Quadrant I (3, 4), (5, 4), (5, 8), and (3, 8).

In short, you rotate the rectangle and then translate it to determine the congruence with the rectangle in Quadrant I.