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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?

(1 point)
translate by (
)

in numbers

Answer 1

To determine the next step after rotating the rectangle in Quadrant III (the first rectangle) 90 degrees clockwise about the origin, we need to examine where the rectangle has moved to after the rotation.

The first rectangle has vertices at:

• (-5, -4)
• (-1, -4)
• (-1, -6)
• (-5, -6)

When rotated 90 degrees clockwise about the origin, the coordinates of these points will transform as follows:

• (-5, -4) -> (-(-4), 5) = (4, -5)
• (-1, -4) -> (-(-4), 1) = (4, -1)
• (-1, -6) -> (-(-6), 1) = (6, -1)
• (-5, -6) -> (-(-6), 5) = (6, -5)

Now, after the rotation, you should translate the newly positioned rectangle (with vertices at (4, -5), (4, -1), (6, -1), (6, -5)) to align it with the rectangle in Quadrant I (the second rectangle) defined by:

• (3, 4)
• (5, 4)
• (5, 8)
• (3, 8)

To do this translation, you need to find the difference between the x-coordinates and the y-coordinates of a corresponding point in Quadrant I and the newly positioned rectangle in Quadrant III after rotation. Picking, for example, the point (4, -5) from the rotated rectangle and (3, 4) from the target rectangle, the translation vector can be determined as follows:

For x-coordinate:

• Target x (3) - Rotated x (4) = 3 - 4 = -1

For y-coordinate:

• Target y (4) - Rotated y (-5) = 4 - (-5) = 4 + 5 = 9

Thus, the translation vector needed is (-1, 9).

Hence, the answer is:

translate by (-1, 9)