Let's analyze the given situation step by step to determine which reflection produces the required endpoints.
1. Original coordinates:
- Endpoint 1: [tex]\((-4, -6)\)[/tex]
- Endpoint 2: [tex]\((-6, 4)\)[/tex]
2. Reflections:
- Across the [tex]\(x\)[/tex]-axis: The [tex]\(y\)[/tex]-coordinates change their signs, the [tex]\(x\)[/tex]-coordinates remain the same.
- Endpoint 1: [tex]\((-4, -(-6)) = (-4, 6)\)[/tex]
- Endpoint 2: [tex]\((-6, -(4)) = (-6, -4)\)[/tex]
- Across the [tex]\(y\)[/tex]-axis: The [tex]\(x\)[/tex]-coordinates change their signs, the [tex]\(y\)[/tex]-coordinates remain the same.
- Endpoint 1: [tex]\((-(-4), -6) = (4, -6)\)[/tex]
- Endpoint 2: [tex]\((-(-6), 4) = (6, 4)\)[/tex]
- Across the line [tex]\(y = x\)[/tex]: The [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates are swapped.
- Endpoint 1: [tex]\((-6, -4)\)[/tex]
- Endpoint 2: [tex]\((4, -6)\)[/tex]
- Across the line [tex]\(y = -x\)[/tex]: Each coordinate swaps places and changes signs.
- Endpoint 1: [tex]\((6, 4)\)[/tex]
- Endpoint 2: [tex]\((-4, -6)\)[/tex]
3. Identifying the correct reflection:
To check, we see the given final endpoints are [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
When we reflect the original endpoints [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] across the [tex]\(y\)[/tex]-axis, we get:
- Endpoint 1: [tex]\((4, -6)\)[/tex]
- Endpoint 2: [tex]\((6, 4)\)[/tex]
These match the desired endpoints exactly.
Conclusion:
The correct reflection that produces endpoints at [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] is the reflection across the [tex]\(y\)[/tex]-axis.
