Use the image to answer the question

Question

An illustration of a coordinate plane of 4 quadrants shows x and y axes ranging from negative 8 to 8 in increments of 1. A rectangle upper A, upper B, upper C, upper D is formed by solid lines at upper A left parenthesis 2 comma 1 right parenthesis, upper B left parenthesis 7 comma 1 right parenthesis, upper C left parenthesis 2 comma 2 right parenthesis, and upper D left parenthesis 7 comma 2 right parenthesis. A rectangle upper A prime, upper B prime, upper C prime, upper D prime is formed by dotted lines at upper A prime left parenthesis 1 comma negative 2 right parenthesis, upper B prime left parenthesis 1 comma negative 7 right parenthesis, upper C prime left parenthesis 2 comma negative 2 right parenthesis, upper D prime left parenthesis 2 comma negative 7 right parenthesis.

What single transformation proves congruence between the two figures? Enter 1 for translation, 2 for reflection, 3 for rotation, or 4 for dilation.

(1 point)
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Answer 1

To determine the transformation that proves congruence between rectangle ABCD and rectangle A'B'C'D', we can analyze their positions on the coordinate plane.

1. Rectangle ABCD:

   • A(2, 1)
   • B(7, 1)
   • C(2, 2)
   • D(7, 2)

2. Rectangle A'B'C'D':

   • A'(1, -2)
   • B'(1, -7)
   • C'(2, -2)
   • D'(2, -7)

Now, notice that rectangle ABCD lies entirely in the first quadrant, while rectangle A'B'C'D' lies in the fourth quadrant.

If we look at the coordinates more closely, both rectangles have the same width and height, but their orientation and position are different.

The transformation that maps points from one rectangle to the other while preserving size and shape must be a reflection across a line.

From the analysis, it appears that:

• A and A' are both at the leftmost vertical line of their respective rectangles but are reflected across the x-axis.
• B and B' are at the rightmost vertical lines of their respective rectangles.
• The horizontal lengths are the same (5 units wide) and the vertical heights (1 unit tall).

Thus, the transformation that proves congruence between the two figures is reflection.

Therefore, the answer is:

2 for reflection.