Question

Trapezoid ABCD coordinates: A(3,1), B(7,1) C(6,4)D(4,4)
Trapezoid A'B'C'D' coordinates: A'(-1,3) B'(-1,7) C'(-4,6) 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=z
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)
Which statement is correct?

Answer 1

To determine which statement is correct about the congruency of trapezoids ABCD and A'B'C'D', let's analyze the coordinates:

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

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

To compare the coordinates, let's test each statement to see if they correctly describe a transformation that leads from ABCD to A'B'C'D'.

**Statement #1:** **Reflection over the line y = z.**
When reflecting a point (x, y) over the line y = x, the new coordinates become (y, x). Applying this transformation:
- A(3, 1) becomes (1, 3)
- B(7, 1) becomes (1, 7)
- C(6, 4) becomes (4, 6)
- D(4, 4) becomes (4, 4)
This does not match A', B', C', and D'.

**Statement #2:** **Translation left 4 units and up 2 units.**
Translating points by subtracting 4 from the x-coordinate and adding 2 to the y-coordinate:
- A(3, 1) => A'(-1, 3)
- B(7, 1) => B'(-1, 7)
- C(6, 4) => C'(-4, 6)
- D(4, 4) => D'(-4, 4)
This matches all the points exactly. This statement seems to be correct.

**Statement #3:** **Rotation 270 degrees (counterclockwise).**
Rotating 270 degrees (or 90 degrees clockwise) would not yield the same points as A'B'C'D'.

**Statement #4:** **Rotation 90 degrees (counterclockwise).**
When rotating a point (x, y) 90 degrees counterclockwise, the new coordinates become (-y, x).
Applying to ABCD:
- A(3, 1) becomes (-1, 3)
- B(7, 1) becomes (-1, 7)
- C(6, 4) becomes (-4, 6)
- D(4, 4) becomes (-4, 4)
This matches A', B', C', D'.

Therefore, both **Statement #2 (translation left 4 units and up 2 units)** and **Statement #4 (rotation 90 degrees counterclockwise)** correctly describe transformations that result in congruent trapezoids.

However, **Statement #2** clearly is a simpler and more straightforward way to reach from ABCD to A'B'C'D', as it involves a direct translation rather than a rotation.

So the correct conclusion is that **Statement #2** is one correct explanation for the congruency of the trapezoids.