A coordinate plane with 4 quadrants shows x and y axes ranging from negative 8 to 8 in increments of 1. Three rectangles are formed by lines joining four plotted points each. Two of the rectangles are joined by rays connecting complementary vertices. The coordinates of the plotted points for the rectangle in quadrant 4 are left parenthesis 1 comma negative 3 right parenthesis, left parenthesis 1 comma negative 7 right parenthesis, left parenthesis 4 comma negative 7 right parenthesis, and left parenthesis 4 comma negative 3 right parenthesis. The coordinates for the rectangle in quadrant 2 are left parenthesis negative 2 comma 1 right parenthesis, left parenthesis negative 6 comma 1 right parenthesis, left parenthesis negative 6 comma 4 right parenthesis, and left parenthesis negative 2 comma 4 right parenthesis. The coordinates of the plotted points for the rectangle in quadrant 3 are left parenthesis negative 3 comma negative 1 right parenthesis, left parenthesis negative 7 comma negative 1 right parenthesis, left parenthesis negative 7 comma negative 4 right parenthesis, and left parenthesis negative 3 comma negative 4 right parenthesis. Four rays from each vertex of the rectangle in quadrant 3 connect to the complementary vertices of the rectangle in quadrant 2. A ray connects the point left parenthesis negative 7 comma negative 4 right parenthesis to left parenthesis negative 6 comma 1 right parenthesis. A ray connects the point left parenthesis negative 7 comma negative 1 right parenthesis to left parenthesis negative 6 comma 4 right parenthesis. A ray connects the point left parenthesis negative 3 comma negative 4 right parenthesis to left parenthesis negative 2 comma 1 right parenthesis. A ray connects the point left parenthesis negative 3 comma negative 1 right parenthesis to left parenthesis negative 2 comma 4 right parenthesis. The arrow heads are on the vertices of the rectangle in quadrant 2.

Question

How would you describe the series of transformations that have occurred to move the rectangle in Quadrant IV to the rectangle in Quadrant II to show that the rectangles are congruent?
rotation, then reflection
rotation, then reflection

reflection, then translation
reflection, then translation

translation, then rotation
translation, then rotation

rotation, then translation

Answer 1

To determine how to move the rectangle in Quadrant IV to the rectangle in Quadrant II, let's analyze the coordinates of the rectangles.

The rectangle in Quadrant IV has the following corners:

• A(1, -3)
• B(1, -7)
• C(4, -7)
• D(4, -3)

The rectangle in Quadrant II has these corners:

• E(-2, 1)
• F(-6, 1)
• G(-6, 4)
• H(-2, 4)

Step 1: Identify Transformation Requirements

1. Reflection: The rectangles are in different quadrants, and to align them, the rectangle in Quadrant IV (which has positive x-coordinates and negative y-coordinates) needs to be flipped into Quadrant II (which has negative x-coordinates and positive y-coordinates).

   If we reflect the rectangle over the y-axis, we transform the x-coordinates of the rectangle in Quadrant IV from positive to negative:

   • A(1, -3) → A'(-1, -3)
   • B(1, -7) → B'(-1, -7)
   • C(4, -7) → C'(-4, -7)
   • D(4, -3) → D'(-4, -3)

   However, we find that reflection alone will not place the rectangle in Quadrant II.

2. Translation: After reflection, we need to translate the new coordinates to their correct positions in Quadrant II:

   • A'(-1, -3) needs to move to E(-2, 1)
   • Similarly for points B', C', and D', they need to be translated appropriately.

Conclusion

The movement involves first reflecting the rectangle across the y-axis and then translating it upward into Quadrant II.

Answer

The series of transformations that have occurred to move the rectangle in Quadrant IV to the rectangle in Quadrant II is reflection, then translation.