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To determine which reflection will produce the image with endpoints at [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex] from the original endpoints [tex]\((-4, -6)\)[/tex] and [tex]\((-6, 4)\)[/tex], we need to explore the effect of each type of reflection specified:
1. Reflection across the [tex]\(x\)[/tex]-axis:
The rule for reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis is [tex]\((x, -y)\)[/tex]. Applying this rule to each endpoint:
- For [tex]\((-4, -6)\)[/tex], the reflection is [tex]\((-4, 6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], the reflection is [tex]\((-6, -4)\)[/tex].
Resulting endpoints: [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
The rule for reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis is [tex]\((-x, y)\)[/tex]. Applying this rule to each endpoint:
- For [tex]\((-4, -6)\)[/tex], the reflection is [tex]\((4, -6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], the reflection is [tex]\((6, 4)\)[/tex].
Resulting endpoints: [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
The rule for reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] is [tex]\((y, x)\)[/tex]. Applying this rule to each endpoint:
- For [tex]\((-4, -6)\)[/tex], the reflection is [tex]\((-6, -4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], the reflection is [tex]\((4, -6)\)[/tex].
Resulting endpoints: [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
The rule for reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] is [tex]\((-y, -x)\)[/tex]. Applying this rule to each endpoint:
- For [tex]\((-4, -6)\)[/tex], the reflection is [tex]\((6, 4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], the reflection is [tex]\((-4, -6)\)[/tex].
Resulting endpoints: [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
By comparing the calculated reflection points to the given image endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], we see that the reflected endpoints match exactly with the reflection across the [tex]\(y\)[/tex]-axis.
Therefore, the correct reflection is a reflection of the line segment across the [tex]\(y\)[/tex]-axis.
1. Reflection across the [tex]\(x\)[/tex]-axis:
The rule for reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis is [tex]\((x, -y)\)[/tex]. Applying this rule to each endpoint:
- For [tex]\((-4, -6)\)[/tex], the reflection is [tex]\((-4, 6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], the reflection is [tex]\((-6, -4)\)[/tex].
Resulting endpoints: [tex]\((-4, 6)\)[/tex] and [tex]\((-6, -4)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
The rule for reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis is [tex]\((-x, y)\)[/tex]. Applying this rule to each endpoint:
- For [tex]\((-4, -6)\)[/tex], the reflection is [tex]\((4, -6)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], the reflection is [tex]\((6, 4)\)[/tex].
Resulting endpoints: [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
The rule for reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] is [tex]\((y, x)\)[/tex]. Applying this rule to each endpoint:
- For [tex]\((-4, -6)\)[/tex], the reflection is [tex]\((-6, -4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], the reflection is [tex]\((4, -6)\)[/tex].
Resulting endpoints: [tex]\((-6, -4)\)[/tex] and [tex]\((4, -6)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
The rule for reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] is [tex]\((-y, -x)\)[/tex]. Applying this rule to each endpoint:
- For [tex]\((-4, -6)\)[/tex], the reflection is [tex]\((6, 4)\)[/tex].
- For [tex]\((-6, 4)\)[/tex], the reflection is [tex]\((-4, -6)\)[/tex].
Resulting endpoints: [tex]\((6, 4)\)[/tex] and [tex]\((-4, -6)\)[/tex].
By comparing the calculated reflection points to the given image endpoints [tex]\((4, -6)\)[/tex] and [tex]\((6, 4)\)[/tex], we see that the reflected endpoints match exactly with the reflection across the [tex]\(y\)[/tex]-axis.
Therefore, the correct reflection is a reflection of the line segment across the [tex]\(y\)[/tex]-axis.
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