Question

Which explanation for the congruency of the trapezoids is correct?

Statement #1: The trapezoids are congruent because ABCD
𝐴
𝐵
𝐶
𝐷
was reflected over the line y=x
𝑦
=
𝑥
.

Statement #2: The trapezoids are congruent because ABCD
𝐴
𝐵
𝐶
𝐷
was translated left 4 units and up 2 units.

Statement #3: The trapezoids are congruent because ABCD
𝐴
𝐵
𝐶
𝐷
was rotated 270°
270
°
(counterclockwise).

Statement #4: The trapezoids are congruent because ABCD
𝐴
𝐵
𝐶
𝐷
was rotated 90°
90
°
(counterclockwise).

Answer 1

I can't tell which statement is correct without the picture or the coordinates of the vertices. If you can upload the diagram or give coordinates for A, B, C, D and their image A′, B′, C′, D′ I will pick the correct transformation.

Here are quick tests you can use (very fast with the coordinates):

1. Translation (left 4, up 2)
- Test: x′ = x − 4 and y′ = y + 2 for every vertex.
- If true for all vertices, Statement #2 is correct.

2. Reflection across y = x
- Test: x′ = y and y′ = x for every vertex (i.e., swap coordinates).
- If true for all vertices, Statement #1 is correct.

3. Rotation 90° counterclockwise
- Formula: (x,y) → (−y, x).
- If (x′,y′) = (−y,x) for every vertex, Statement #4 is correct.

4. Rotation 270° counterclockwise (same as 90° clockwise)
- Formula: (x,y) → (y, −x).
- If (x′,y′) = (y,−x) for every vertex, Statement #3 is correct.

A quick shortcut: check one vertex pair—if it fails, that transformation is out. Also note reflections reverse the polygon’s orientation (clockwise ↔ counterclockwise); rotations and translations preserve orientation. If you post the coordinates or an image I’ll apply these tests and give the answer.