To determine which reflection would transform the endpoints of a line segment from [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] to [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], follow these steps:
1. Understand the Transformations:
- Reflection across the [tex]\(x\)[/tex]-axis: Changes the point [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex].
- Reflection across the [tex]\(y\)[/tex]-axis: Changes the point [tex]\((x, y)\)[/tex] to [tex]\((-x, y)\)[/tex].
- Reflection across the line [tex]\(y = x\)[/tex]: Changes the point [tex]\((x, y)\)[/tex] to [tex]\((y, x)\)[/tex].
- Reflection across the line [tex]\(y = -x\)[/tex]: Changes the point [tex]\((x, y)\)[/tex] to [tex]\((-y, -x)\)[/tex].
2. Check each possible reflection:
- Reflection across the [tex]\(x\)[/tex]-axis:
- [tex]\((-4, -6) \rightarrow (-4, 6)\)[/tex]
- [tex]\((-6, 4) \rightarrow (-6, -4)\)[/tex]
- The resulting points are [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex], which do not match the target points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
- Reflection across the [tex]\(y\)[/tex]-axis:
- [tex]\((-4, -6) \rightarrow (4, -6)\)[/tex]
- [tex]\((-6, 4) \rightarrow (6, 4)\)[/tex]
- The resulting points are [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], which match the target points exactly.
- Reflection across the line [tex]\(y = x\)[/tex]:
- [tex]\((-4, -6) \rightarrow (-6, -4)\)[/tex]
- [tex]\((-6, 4) \rightarrow (4, -6)\)[/tex]
- The resulting points are [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex], which do not match the target points.
- Reflection across the line [tex]\(y = -x\)[/tex]:
- [tex]\((-4, -6) \rightarrow (6, 4)\)[/tex]
- [tex]\((-6, 4) \rightarrow (-4, -6)\)[/tex]
- The resulting points are [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex], which do not match the target points.
3. Conclusion:
Only the reflection across the [tex]\(y\)[/tex]-axis produces the exact target points [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex]. Thus, the correct reflection is:
A reflection of the line segment across the [tex]\(y\)[/tex]-axis.
Therefore, the solution to the problem is the reflection across the [tex]\(y\)[/tex]-axis.
