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A line segment has endpoints at [tex]$(-4, -6)$[/tex] and [tex]$(-6, 4)$[/tex]. Which reflection will produce an image with endpoints at [tex]$(4, -6)$[/tex] and [tex]$(6, 4)$[/tex]?

A. A reflection of the line segment across the [tex]$x$[/tex]-axis
B. A reflection of the line segment across the [tex]$y$[/tex]-axis
C. A reflection of the line segment across the line [tex]$y = x$[/tex]
D. A reflection of the line segment across the line [tex]$y = -x$[/tex]

Sagot :

GinnyAnswer
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.
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