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.