Question

In the coordinate plane below, figure ABCD is a parallelogram, and side AD and side BC are horizontal line segments.



If parallelogram ABCD is reflected across the x-axis to form figure A'B'C'D', which of the following statement are true? Select all true statements.

Side B'C' is parallel to side BC
Side C'D' is parallel to side CD
Side A'B' is parallel to side AB
Side A'B' is parallel to side C'D'
Side A'D' is parallel to side B'C'
Side A'B' is parallel to side A'D'

Answer 1

When a parallelogram ABCD is reflected across the x-axis, the coordinates of the vertices change as follows:

- A(x, y) → A'(x, -y)
- B(x, y) → B'(x, -y)
- C(x, y) → C'(x, -y)
- D(x, y) → D'(x, -y)

Given the properties of a parallelogram (opposite sides are equal and parallel), let’s analyze the statements:

1. **Side B'C' is parallel to side BC**
This is true. Since reflection preserves parallelism, B'C' will be parallel to BC.

2. **Side C'D' is parallel to side CD**
This is true. C'D' will be parallel to CD due to the same reason.

3. **Side A'B' is parallel to side AB**
This is true. A'B' will be parallel to AB because reflection keeps parallel lines parallel.

4. **Side A'B' is parallel to side C'D'**
This is true. Since both A'B' and C'D' are parallel to their respective original sides (AB and CD) and AB is parallel to CD in parallelogram ABCD, A'B' will be parallel to C'D'.

5. **Side A'D' is parallel to side B'C'**
This is true. Similar to the earlier reasoning, A'D' is parallel to AD (by properties of parallelograms) and B'C' is parallel to BC; and since AD is parallel to BC, A'D' will be parallel to B'C'.

6. **Side A'B' is parallel to side A'D'**
This is true. Since A'B' is parallel to AB and A'D' is parallel to AD, and in a parallelogram, AB is parallel to AD, then A'B' will be parallel to A'D'.

Based on this reasoning, **all of the statements are true**:

- Side B'C' is parallel to side BC.
- Side C'D' is parallel to side CD.
- Side A'B' is parallel to side AB.
- Side A'B' is parallel to side C'D'.
- Side A'D' is parallel to side B'C'.
- Side A'B' is parallel to side A'D'.