Congruent Figures Practice

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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 (
)

Answer 1

To determine the next step after rotating the first rectangle (in Quadrant III) 90 degrees clockwise about the origin, you need to translate the resulting rectangle to align it with the one in Quadrant I.

The original coordinates of the first rectangle in Quadrant III are:

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

After a 90-degree clockwise rotation around the origin, these points will transform to:

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

Next, to translate this rotated rectangle to overlap with the rectangle in Quadrant I, you will need to move it up vertically to the positive y-values. The rectangle in Quadrant I has its coordinates as follows:

• \( (3, 4) \)
• \( (5, 4) \)
• \( (5, 8) \)
• \( (3, 8) \)

The translation will be determined by how far you need to move the new coordinates of the rotated rectangle up to match the y-values in Quadrant I.

From \( (4, -5) \) which is part of the rotated rectangle, to match up with the expected y-values that go from 4 to 8, you'll translate it by:

• Moving 5 units up to raise the y-values from -5 to 0.
• Moving 4 more units up will raise the y-values from 0 to 4.

Thus, the total translation is to move up 9 units.

So the translation to perform after the rotation is:

Translate by (0, 9).