Question

△ABC
is reflected over the line x=2
to create △A′B′C′
, which is then reflected over the line y=−2
. Which of the following could describe this composition transformation using a single transformation?(1 point)
Responses

a rotation about the intersection of the two perpendicular lines
a rotation about the intersection of the two perpendicular lines

a translation in the same direction as the reflections
a translation in the same direction as the reflections

a reflection over the line
a reflection over the line

a dilation by a certain factor

Answer 1

To analyze the composition of the transformations given in the problem, we first break down the steps:

1. **Reflection over the line \( x = 2 \)**: When a point \( (x, y) \) is reflected over the vertical line \( x = 2 \), the new coordinates of the reflected point will be \( (4 - x, y) \).

2. **Reflection over the line \( y = -2 \)**: When a point \( (x, y) \) is reflected over the horizontal line \( y = -2 \), the new coordinates of the reflected point will be \( (x, -4 - y) \).

To analyze the overall transformation, let's consider a point \( P(x, y) \):

- After the first reflection over \( x = 2 \):
\[
P' = (4 - x, y)
\]

- Then, reflecting \( P' \) over \( y = -2 \):
\[
P'' = (4 - x, -4 - y)
\]

We can analyze how this transformation changes the coordinates.

Now, let's define:
- \( x' = 4 - x \)
- \( y' = -4 - y \)

From here, we can express the final point \( P'' \) in terms of the original point \( P \):
- \( P'' = (4 - x, -4 - y) \)

### To find a single transformation that summarizes these two reflections:

1. The first transformation changes \( x \) to \( 4 - x \).
2. The second transformation changes \( y \) to \( -4 - y \).

Ultimately, this can be thought of as a **rotation**. Specifically, a composition of two reflections across two orthogonal lines results in a rotation about the intersection point of those two lines.

### Conclusion:
The combination of reflections across two perpendicular lines results in a rotation. Therefore, the best description of the composition transformation using a single transformation is:

**a rotation about the intersection of the two perpendicular lines.**