Use the coordinates to answer the question.

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).