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To determine which reflection will transform the endpoints [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] of a line segment to [tex]\( (-4, 1) \)[/tex] and [tex]\((-1, -4)\)[/tex], let's analyze each type of reflection step by step.
1. Reflection across the [tex]\(x\)[/tex]-axis:
- For this type of reflection, each point [tex]\((x, y)\)[/tex] is transformed to [tex]\((x, -y)\)[/tex].
- Therefore, reflecting [tex]\((-1, 4)\)[/tex] gives us [tex]\((-1, -4)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] gives us [tex]\((4, -1)\)[/tex].
- Result: [tex]\((-1, -4)\)[/tex] and [tex]\((4, -1)\)[/tex].
- This does not match the desired endpoints, so reflecting across the [tex]\(x\)[/tex]-axis is not the correct reflection.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- For this reflection, each point [tex]\((x, y)\)[/tex] is transformed to [tex]\((-x, y)\)[/tex].
- Therefore, reflecting [tex]\((-1, 4)\)[/tex] gives us [tex]\((1, 4)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] gives us [tex]\((-4, 1)\)[/tex].
- Result: [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex].
- This does not match the desired endpoints, so reflecting across the [tex]\(y\)[/tex]-axis is not the correct reflection.
3. Reflection across the line [tex]\(y=x\)[/tex]:
- For this reflection, each point [tex]\((x, y)\)[/tex] is transformed to [tex]\((y, x)\)[/tex].
- Therefore, reflecting [tex]\((-1, 4)\)[/tex] gives us [tex]\((4, -1)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] gives us [tex]\((1, 4)\)[/tex].
- Result: [tex]\((4, -1)\)[/tex] and [tex]\((1, 4)\)[/tex].
- This does not match the desired endpoints, so reflecting across the line [tex]\(y=x\)[/tex] is not the correct reflection.
4. Reflection across the line [tex]\(y=-x\)[/tex]:
- For this reflection, each point [tex]\((x, y)\)[/tex] is transformed to [tex]\((-y, -x)\)[/tex].
- Therefore, reflecting [tex]\((-1, 4)\)[/tex] gives us [tex]\((-4, 1)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] gives us [tex]\((-1, -4)\)[/tex].
- Result: [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
- This matches the desired endpoints precisely.
Hence, the correct reflection that will transform the endpoints [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] to [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] is the reflection across the line [tex]\(y = -x\)[/tex].
Final Answer: A reflection of the line segment across the line [tex]\(y = -x\)[/tex].
1. Reflection across the [tex]\(x\)[/tex]-axis:
- For this type of reflection, each point [tex]\((x, y)\)[/tex] is transformed to [tex]\((x, -y)\)[/tex].
- Therefore, reflecting [tex]\((-1, 4)\)[/tex] gives us [tex]\((-1, -4)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] gives us [tex]\((4, -1)\)[/tex].
- Result: [tex]\((-1, -4)\)[/tex] and [tex]\((4, -1)\)[/tex].
- This does not match the desired endpoints, so reflecting across the [tex]\(x\)[/tex]-axis is not the correct reflection.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- For this reflection, each point [tex]\((x, y)\)[/tex] is transformed to [tex]\((-x, y)\)[/tex].
- Therefore, reflecting [tex]\((-1, 4)\)[/tex] gives us [tex]\((1, 4)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] gives us [tex]\((-4, 1)\)[/tex].
- Result: [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex].
- This does not match the desired endpoints, so reflecting across the [tex]\(y\)[/tex]-axis is not the correct reflection.
3. Reflection across the line [tex]\(y=x\)[/tex]:
- For this reflection, each point [tex]\((x, y)\)[/tex] is transformed to [tex]\((y, x)\)[/tex].
- Therefore, reflecting [tex]\((-1, 4)\)[/tex] gives us [tex]\((4, -1)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] gives us [tex]\((1, 4)\)[/tex].
- Result: [tex]\((4, -1)\)[/tex] and [tex]\((1, 4)\)[/tex].
- This does not match the desired endpoints, so reflecting across the line [tex]\(y=x\)[/tex] is not the correct reflection.
4. Reflection across the line [tex]\(y=-x\)[/tex]:
- For this reflection, each point [tex]\((x, y)\)[/tex] is transformed to [tex]\((-y, -x)\)[/tex].
- Therefore, reflecting [tex]\((-1, 4)\)[/tex] gives us [tex]\((-4, 1)\)[/tex].
- Reflecting [tex]\((4, 1)\)[/tex] gives us [tex]\((-1, -4)\)[/tex].
- Result: [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex].
- This matches the desired endpoints precisely.
Hence, the correct reflection that will transform the endpoints [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] to [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] is the reflection across the line [tex]\(y = -x\)[/tex].
Final Answer: A reflection of the line segment across the line [tex]\(y = -x\)[/tex].
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