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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° (counterclockwise). Statement #4: The trapezoids are congruent because ABCD was rotated 90° (counterclockwise). (1 point) Statement #___ is correct.

3 answers

To determine which statement about the congruency of the trapezoids is correct, you need to consider the effects of each transformation mentioned in the statements:

  1. Reflection over the line \(y=x\) would produce a congruent figure, but it will not change the shape or size of the trapezoid, just its orientation.

  2. Translation does not change the shape or size of the figure. A translation left 4 units and up 2 units will create a congruent trapezoid.

  3. Rotation (whether 270° counterclockwise or 90° counterclockwise) will also not change the shape or size of the trapezoid. Both rotations will produce a congruent figure, as rotation preserves congruency.

Since all transformations stated in the explanations will produce congruent figures, the answer could potentially involve more than one statement being correct. However, the most general statement that can apply to all congruent transformations is translation, reflection, and rotation.

Without more specific geometrical context (like the actual trapezoids), it is challenging to determine which specific transformation made the trapezoids congruent.

However, since the question requests a specific correct statement, if I had to choose one based on the simplicity of explanation, I would go with:

Statement #2 is correct: "The trapezoids are congruent because ABCD was translated left 4 units and up 2 units."

This is because translation is a straightforward method to confirm congruency without changing orientation.

The coordinates of ABCD are, (3,1) (7,1) (6,4) (4,4). Coordinates of A' B' C' D' are, (-1,3) (-1,7) (-4,6) (-4,4)

To determine which statement about the congruency of the trapezoids ABCD and A'B'C'D' is correct, let’s examine the given coordinates for both trapezoids.

Coordinates for ABCD:

  • A(3, 1)
  • B(7, 1)
  • C(6, 4)
  • D(4, 4)

Coordinates for A'B'C'D':

  • A'(-1, 3)
  • B'(-1, 7)
  • C'(-4, 6)
  • D'(-4, 4)

Analyzing Transformations:

  1. Check for Reflection over the line \(y = x\): When reflecting a point \((x, y)\) over the line \(y = x\), the coordinates become \((y, x)\).

    • A(3, 1) → (1, 3) (not A')
    • B(7, 1) → (1, 7) (not B')
    • C(6, 4) → (4, 6) (matches C')
    • D(4, 4) → (4, 4) (matches D')

    Since not all points reflect properly, Statement #1 is incorrect.

  2. Check for Translation: To check if there was a translation, find the differences in coordinates:

    • From A(3, 1) to A'(-1, 3): \[ \Delta x = -1 - 3 = -4 \] \[ \Delta y = 3 - 1 = +2 \]

    • From B(7, 1) to B'(-1, 7): \[ \Delta x = -1 - 7 = -8 \] \[ \Delta y = 7 - 1 = +6 \]

    Since the translation is not consistent for all points, Statement #2 is incorrect.

  3. Check for Rotation: To see if rotation could produce congruent figures, we need to verify whether a consistent angle and center of rotation can transform points correctly. Without calculation, it isn't straightforward to confirm this statement conclusively without the center of rotation.

  4. Check for 90° Counterclockwise rotation: A 90° counterclockwise rotation around the origin transforms \((x, y)\) to \((-y, x)\).

    • A(3, 1) → (-1, 3) (matches A')
    • B(7, 1) → (-1, 7) (matches B')
    • C(6, 4) → (-4, 6) (matches C')
    • D(4, 4) → (-4, 4) (matches D')

    It appears that these transformations hold true for all points, indicating a valid 90° counter-clockwise rotation.

Final Conclusion:

  • Statement #4 is correct: "The trapezoids are congruent because ABCD was rotated 90° (counterclockwise)."