Question
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 dilation by a certain factor
a dilation by a certain factor
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 rotation about the intersection of the two perpendicular lines
△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 dilation by a certain factor
a dilation by a certain factor
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 rotation about the intersection of the two perpendicular lines
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To analyze the composition of the two reflections described (first over the line \( x = 2 \) and then over the line \( y = -2 \)), we can consider how each reflection affects a point.
1. **Reflecting over \( x = 2 \)**: This will move a point \( A(x, y) \) to \( A'(4 - x, y) \). The \( x \)-coordinate is transformed to its reflection across the line \( x = 2 \).
2. **Reflecting over \( y = -2 \)**: Then, reflecting \( A'(4 - x, y) \) over \( y = -2 \) involves changing the \( y \)-coordinate. The new \( y'\)-coordinate will be \( -2 - (y - (-2)) = -2 + 2 - y = -y \). Thus, the final point would be:
\[
A''(4 - x, -y)
\]
After performing both reflections, we can see how this composition of transformations can be analyzed:
- Reflecting over the vertical line \( x = 2 \) and then the horizontal line \( y = -2 \) results in a transformation that can be interpreted as a **rotation** around the point where these two lines intersect. This point of intersection is \( (2, -2) \). The transformation overall is equivalent to a rotation of 180 degrees about this intersection point.
Based on this reasoning, the correct description of the composition transformation is:
**a rotation about the intersection of the two perpendicular lines.**
1. **Reflecting over \( x = 2 \)**: This will move a point \( A(x, y) \) to \( A'(4 - x, y) \). The \( x \)-coordinate is transformed to its reflection across the line \( x = 2 \).
2. **Reflecting over \( y = -2 \)**: Then, reflecting \( A'(4 - x, y) \) over \( y = -2 \) involves changing the \( y \)-coordinate. The new \( y'\)-coordinate will be \( -2 - (y - (-2)) = -2 + 2 - y = -y \). Thus, the final point would be:
\[
A''(4 - x, -y)
\]
After performing both reflections, we can see how this composition of transformations can be analyzed:
- Reflecting over the vertical line \( x = 2 \) and then the horizontal line \( y = -2 \) results in a transformation that can be interpreted as a **rotation** around the point where these two lines intersect. This point of intersection is \( (2, -2) \). The transformation overall is equivalent to a rotation of 180 degrees about this intersection point.
Based on this reasoning, the correct description of the composition transformation is:
**a rotation about the intersection of the two perpendicular lines.**
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.