Question

Use the coordinates to answer the question.
Trapezoid ABCD:
Point A: (3, 1)
Point B: (7, 1)
Point C: (6, 4)
Point D: (4, 4)

Trapezoid A'B'C'D':
Point A': (-1, 3)
Point B': (-1, 7)
Point C': (-4, 6)
Point D': (-4, 4)

Which explanation for the congruency of the trapezoids is correct?
Statement #1: The trapezoids are congruent because ABCD was reflected over the line y = x.
Statement #2: The trapezoids are congruent because ABCD was translated left 4 units and up 2 units.
Statement #3: The trapezoids are congruent because ABCD was rotated 270 degrees (counterclockwise).
Statement #4: The trapezoids are congruent because ABCD was rotated 90 degrees (counterclockwise).

Answer 1

To determine which statement about the congruency of trapezoids ABCD and A'B'C'D' is correct, we need to analyze the coordinates of the two trapezoids.

### Coordinates of Trapezoid ABCD:
- Point A: (3, 1)
- Point B: (7, 1)
- Point C: (6, 4)
- Point D: (4, 4)

### Coordinates of Trapezoid A'B'C'D':
- Point A': (-1, 3)
- Point B': (-1, 7)
- Point C': (-4, 6)
- Point D': (-4, 4)

### Analysis of the Statements:

1. **Statement #1: The trapezoids are congruent because ABCD was reflected over the line y = x.**
- Reflecting over the line y = x swaps the x and y coordinates of each point. This means:
- A would go from (3, 1) to (1, 3)
- B would go from (7, 1) to (1, 7)
- C would go from (6, 4) to (4, 6)
- D would go from (4, 4) to (4, 4)
- The resulting points do not match A', B', C', and D', therefore this statement is **not correct**.

2. **Statement #2: The trapezoids are congruent because ABCD was translated left 4 units and up 2 units.**
- Translating left 4 units and up 2 units means adjusting the coordinates as follows:
- A goes from (3, 1) to (-1, 3)
- B goes from (7, 1) to (-1, 7)
- C goes from (6, 4) to (2, 6)
- D goes from (4, 4) to (0, 6)
- The resulting points do not match A', B', C', and D', therefore this statement is **not correct**.

3. **Statement #3: The trapezoids are congruent because ABCD was rotated 270 degrees (counterclockwise).**
- To rotate a point 270 degrees counterclockwise, we can use the transformation (x, y) ➝ (y, -x):
- A would go from (3, 1) to (1, -3)
- B would go from (7, 1) to (1, -7)
- C would go from (6, 4) to (4, -6)
- D would go from (4, 4) to (4, -4)
- The resulting points do not match A', B', C', and D', therefore this statement is **not correct**.

4. **Statement #4: The trapezoids are congruent because ABCD was rotated 90 degrees (counterclockwise).**
- To rotate a point 90 degrees counterclockwise, we can use the transformation (x, y) ➝ (-y, x):
- A would go from (3, 1) to (-1, 3)
- B would go from (7, 1) to (-1, 7)
- C would go from (6, 4) to (-4, 6)
- D would go from (4, 4) to (-4, 4)
- The resulting points match A', B', C', and D', therefore this statement is **correct**.

### Conclusion:
The correct explanation for the congruency of trapezoids ABCD and A'B'C'D' is **Statement #4: The trapezoids are congruent because ABCD was rotated 90 degrees (counterclockwise).**