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 reflection over the line
a reflection over the line
a dilation by a certain factor
a dilation by a certain factor
a translation in the same direction as the reflections
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 reflection over the line
a reflection over the line
a dilation by a certain factor
a dilation by a certain factor
a translation in the same direction as the reflections
Answers
GPT-4o mini
To analyze the composition of the transformations described, let's break it down:
1. **Reflection over the line \(x=2\)**: This reflection flips points across the vertical line \(x=2\). Each point A(x, y) transforms to A'(4-x, y).
2. **Reflection over the line \(y=-2\)**: This second reflection flips points across the horizontal line \(y=-2\). Transforming A'(4-x, y) results in A''(4-x, -4-y).
Now, we have two reflections that can be treated as a single transformation. The composition of two reflections over two intersecting lines results in a rotation around the point of intersection of those two lines.
- The intersection point of the lines \(x=2\) and \(y=-2\) is point \( (2, -2) \).
Thus, the overall transformation that describes the combination of these two reflections is a rotation about the point of intersection of the two lines.
Therefore, the correct response is:
**a rotation about the intersection of the two perpendicular lines**.
1. **Reflection over the line \(x=2\)**: This reflection flips points across the vertical line \(x=2\). Each point A(x, y) transforms to A'(4-x, y).
2. **Reflection over the line \(y=-2\)**: This second reflection flips points across the horizontal line \(y=-2\). Transforming A'(4-x, y) results in A''(4-x, -4-y).
Now, we have two reflections that can be treated as a single transformation. The composition of two reflections over two intersecting lines results in a rotation around the point of intersection of those two lines.
- The intersection point of the lines \(x=2\) and \(y=-2\) is point \( (2, -2) \).
Thus, the overall transformation that describes the combination of these two reflections is a rotation about the point of intersection of the two lines.
Therefore, the correct response is:
**a rotation about the intersection of the two perpendicular lines**.