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Question

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 8 to 8 in increments of 1. Two 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 1 comma negative 2 right parenthesis, left parenthesis negative 3 comma negative 2 right parenthesis, left parenthesis negative 3 comma negative 7 right parenthesis, and left parenthesis negative 1 comma negative 7 right parenthesis. The coordinates for the second rectangle joined by lines made of dashes and dots are left parenthesis 2 comma 3 right parenthesis, left parenthesis 7 comma 3 right parenthesis, left parenthesis 7 comma 5 right parenthesis, and left parenthesis 2 comma 5 right parenthesis.

To show that the rectangle in Quadrant III is congruent to the rectangle in Quadrant I, Darnell rotates the first rectangle 90 degrees clockwise into Quadrant II. What is his next step?

(1 point)

(___, ___)

Answer 1

To determine the next step after rotating the rectangle in Quadrant III 90 degrees clockwise into Quadrant II, we need to identify the new coordinates of the rotated rectangle.

The original coordinates of the rectangle in Quadrant III are:

• (-1, -2)
• (-3, -2)
• (-3, -7)
• (-1, -7)

When Darnell rotates these points 90 degrees clockwise, the transformation for a point (x, y) to (y, -x) applies. Therefore, we can transform each point as follows:

1. (-1, -2) becomes (-2, 1)
2. (-3, -2) becomes (-2, 3)
3. (-3, -7) becomes (-7, 3)
4. (-1, -7) becomes (-7, 1)

After performing the rotation, Darnell now has the new coordinates of the rectangle in Quadrant II:

• (-2, 1)
• (-2, 3)
• (-7, 3)
• (-7, 1)

The next step to demonstrate that these two rectangles are congruent would be to translate the rectangle in Quadrant II to overlay it with the rectangle in Quadrant I.

To do this, we need to translate the coordinates from Quadrant II into the position of the rectangle in Quadrant I. The rectangle in Quadrant I has the coordinates:

• (2, 3)
• (7, 3)
• (7, 5)
• (2, 5)

The bottom-left corner of the rectangle in Quadrant I is at (2, 3), while the bottom-left corner in the translated position from Quadrant II is at (-7, 1).

To move the rectangle from (-7, 1) to (2, 3), we can find the translation vector. The x-coordinate change from -7 to 2 is +9, and the y-coordinate change from 1 to 3 is +2.

Thus, the next step for Darnell is to translate the rectangle by the vector (9, 2).

The final answer for the next step after Darnell rotates the rectangle is:

(9, 2)