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.

Answer 1

To determine the transformation that proves congruence between rectangle ABCD and rectangle A'B'C'D', we can analyze the positions of their corresponding vertices.

- Rectangle ABCD has vertices:
A(2, 1)
B(7, 1)
C(2, 2)
D(7, 2)

- Rectangle A'B'C'D' has vertices:
A'(1, -2)
B'(1, -7)
C'(2, -2)
D'(2, -7)

The differences in the coordinates suggest that the rectangles might be related by a reflection. When we observe the y-coordinates of A and A', or B and B', we notice that the y-coordinates have opposite signs while the x-coordinates stay consistent for points A and A' (2 to 1 and 1 to 2) but with a shift in the y-direction.

To find a specific transformation, if we reflect rectangle ABCD across the line y = 0 (the x-axis), we see that:
- A(2, 1) becomes A'(2, -1)
- B(7, 1) becomes B'(7, -1)
- C(2, 2) becomes C'(2, -2)
- D(7, 2) becomes D'(7, -2)

However, for our specific points, we can conclude that translating each of the vertices down by 3 units (i.e., subtracting 3 from the y-coordinates of A, B, C, and D) will yield the coordinates of A', B', C', and D'.

Thus, the transformation required to show the congruence between ABCD and A'B'C'D' is a combination of translation followed by reflection over the x-axis.

Overall, the simplist transformation that is evident here is actually a reflection, since the y-coordinates merely switch signs. Therefore, the answer is:

**2 for reflection**.