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To determine which reflection will result in the line segment with endpoints at [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] from the original endpoints [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex], we need to examine the effect of each type of reflection on the given endpoints:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- When we reflect a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis, the new coordinates become [tex]\((x, -y)\)[/tex].
- Applying this to our original points:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((-1, -4)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((4, -1)\)[/tex]
- The resulting points are [tex]\((-1, -4)\)[/tex] and [tex]\((4, -1)\)[/tex], which does not match our target endpoints.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- When we reflect a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis, the new coordinates become [tex]\((-x, y)\)[/tex].
- Applying this to our original points:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((1, 4)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((-4, 1)\)[/tex]
- The resulting points are [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex], which does not match our target endpoints.
3. Reflection across the line [tex]\(y = x\)[/tex]:
- When we reflect a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex], the new coordinates become [tex]\((y, x)\)[/tex].
- Applying this to our original points:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((4, -1)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((1, 4)\)[/tex]
- The resulting points are [tex]\((4, -1)\)[/tex] and [tex]\((1, 4)\)[/tex], which does not match our target endpoints.
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When we reflect a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex], the new coordinates become [tex]\((-y, -x)\)[/tex].
- Applying this to our original points:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((-4, 1)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((-1, -4)\)[/tex]
- The resulting points are [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex], which exactly match our target endpoints.
From the above reflections, the only reflection that produces endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] from the original endpoints [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] is the reflection across the line [tex]\(y = -x\)[/tex].
Therefore, the correct answer is:
- A reflection of the line segment across the line [tex]\(y = -x\)[/tex].
1. Reflection across the [tex]\(x\)[/tex]-axis:
- When we reflect a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis, the new coordinates become [tex]\((x, -y)\)[/tex].
- Applying this to our original points:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((-1, -4)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((4, -1)\)[/tex]
- The resulting points are [tex]\((-1, -4)\)[/tex] and [tex]\((4, -1)\)[/tex], which does not match our target endpoints.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- When we reflect a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis, the new coordinates become [tex]\((-x, y)\)[/tex].
- Applying this to our original points:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((1, 4)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((-4, 1)\)[/tex]
- The resulting points are [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex], which does not match our target endpoints.
3. Reflection across the line [tex]\(y = x\)[/tex]:
- When we reflect a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex], the new coordinates become [tex]\((y, x)\)[/tex].
- Applying this to our original points:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((4, -1)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((1, 4)\)[/tex]
- The resulting points are [tex]\((4, -1)\)[/tex] and [tex]\((1, 4)\)[/tex], which does not match our target endpoints.
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- When we reflect a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex], the new coordinates become [tex]\((-y, -x)\)[/tex].
- Applying this to our original points:
- [tex]\((-1, 4)\)[/tex] becomes [tex]\((-4, 1)\)[/tex]
- [tex]\((4, 1)\)[/tex] becomes [tex]\((-1, -4)\)[/tex]
- The resulting points are [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex], which exactly match our target endpoints.
From the above reflections, the only reflection that produces endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] from the original endpoints [tex]\((-1, 4)\)[/tex] and [tex]\((4, 1)\)[/tex] is the reflection across the line [tex]\(y = -x\)[/tex].
Therefore, the correct answer is:
- A reflection of the line segment across the line [tex]\(y = -x\)[/tex].
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