To determine which reflection will transform the endpoints of the line segment from [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] to [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], we need to consider the effect each type of reflection has on the coordinates of points.
1. Reflection across the x-axis:
For a point [tex]\((x, y)\)[/tex], reflection across the x-axis transforms the point to [tex]\((x, -y)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the x-axis:
[tex]\[ (-4, -6) \rightarrow (-4, 6) \][/tex]
[tex]\[ (-6, 4) \rightarrow (-6, -4) \][/tex]
These reflected points, [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex], do not match [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
2. Reflection across the y-axis:
For a point [tex]\((x, y)\)[/tex], reflection across the y-axis transforms the point to [tex]\((-x, y)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the y-axis:
[tex]\[ (-4, -6) \rightarrow (4, -6) \][/tex]
[tex]\[ (-6, 4) \rightarrow (6, 4) \][/tex]
These reflected points, [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], match the desired endpoints.
3. Reflection across the line [tex]\(y = x\)[/tex]:
For a point [tex]\((x, y)\)[/tex], reflection across the line [tex]\(y = x\)[/tex] transforms the point to [tex]\((y, x)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the line [tex]\(y = x\)[/tex]:
[tex]\[ (-4, -6) \rightarrow (-6, -4) \][/tex]
[tex]\[ (-6, 4) \rightarrow (4, -6) \][/tex]
These reflected points, [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex], do not match [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
For a point [tex]\((x, y)\)[/tex], reflection across the line [tex]\(y = -x\)[/tex] transforms the point to [tex]\((-y, -x)\)[/tex].
- Original endpoints: [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex]
- After reflection across the line [tex]\(y = -x\)[/tex]:
[tex]\[ (-4, -6) \rightarrow (6, 4) \][/tex]
[tex]\[ (-6, 4) \rightarrow (-4, -6) \][/tex]
These reflected points, [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex], do not match [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
By considering all the reflections and their effects on the given points, the reflection that produces the endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] from the original endpoints [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex] is the reflection across the y-axis. Therefore, the correct answer is:
A reflection of the line segment across the y-axis.
