Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Get the answers you need quickly and accurately from a dedicated community of experts on our Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To determine which reflection transforms the endpoints of the line segment from [tex]\((-1, 4)\)[/tex] and [tex]\( (4, 1) \)[/tex] to [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex], we need to analyze each possible reflection scenario:
1. Reflection across the [tex]\(x\)[/tex]-axis:
- When reflecting across the [tex]\(x\)[/tex]-axis, the [tex]\(y\)[/tex]-coordinate changes sign, while the [tex]\(x\)[/tex]-coordinate remains the same.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (-1, -4) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (4, -1) \][/tex]
- The reflected points are [tex]\((-1, -4)\)[/tex] and [tex]\( (4, -1) \)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- When reflecting across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate changes sign, while the [tex]\(y\)[/tex]-coordinate remains the same.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (1, 4) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (-4, 1) \][/tex]
- The reflected points are [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
3. Reflection across the line [tex]\( y=x \)[/tex]:
- When reflecting across the line [tex]\(y=x\)[/tex], the coordinates swap their positions.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \rightarrow (4, -1) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \rightarrow (1, 4) \][/tex]
- The reflected points are [tex]\((4, -1)\)[/tex] and [tex]\( (1, 4) \)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
4. Reflection across the line [tex]\( y=-x \)[/tex]:
- When reflecting across the line [tex]\(y=-x\)[/tex], the coordinates swap their positions and change signs.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \rightarrow (-4, 1) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \rightarrow (-1, -4) \][/tex]
- The reflected points are [tex]\((-4, 1)\)[/tex] and [tex]\( (-1, -4) \)[/tex], which match the given reflected points exactly.
Thus, the correct reflection that produces the endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] from the original points [tex]\((-1, 4)\)[/tex] and [tex]\( (4, 1) \)[/tex] is a reflection across the line [tex]\( y = -x \)[/tex].
The correct answer is:
[tex]\[ \boxed{4} \][/tex]
1. Reflection across the [tex]\(x\)[/tex]-axis:
- When reflecting across the [tex]\(x\)[/tex]-axis, the [tex]\(y\)[/tex]-coordinate changes sign, while the [tex]\(x\)[/tex]-coordinate remains the same.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (-1, -4) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (x, -y) \rightarrow (4, -1) \][/tex]
- The reflected points are [tex]\((-1, -4)\)[/tex] and [tex]\( (4, -1) \)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
2. Reflection across the [tex]\(y\)[/tex]-axis:
- When reflecting across the [tex]\(y\)[/tex]-axis, the [tex]\(x\)[/tex]-coordinate changes sign, while the [tex]\(y\)[/tex]-coordinate remains the same.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (1, 4) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-x, y) \rightarrow (-4, 1) \][/tex]
- The reflected points are [tex]\((1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
3. Reflection across the line [tex]\( y=x \)[/tex]:
- When reflecting across the line [tex]\(y=x\)[/tex], the coordinates swap their positions.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \rightarrow (4, -1) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (y, x) \rightarrow (1, 4) \][/tex]
- The reflected points are [tex]\((4, -1)\)[/tex] and [tex]\( (1, 4) \)[/tex], which do not match the given reflected points [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex]. Therefore, this option is incorrect.
4. Reflection across the line [tex]\( y=-x \)[/tex]:
- When reflecting across the line [tex]\(y=-x\)[/tex], the coordinates swap their positions and change signs.
- For the original point [tex]\((-1, 4)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \rightarrow (-4, 1) \][/tex]
- For the original point [tex]\((4, 1)\)[/tex]:
[tex]\[ (x, y) \rightarrow (-y, -x) \rightarrow (-1, -4) \][/tex]
- The reflected points are [tex]\((-4, 1)\)[/tex] and [tex]\( (-1, -4) \)[/tex], which match the given reflected points exactly.
Thus, the correct reflection that produces the endpoints [tex]\((-4, 1)\)[/tex] and [tex]\((-1, -4)\)[/tex] from the original points [tex]\((-1, 4)\)[/tex] and [tex]\( (4, 1) \)[/tex] is a reflection across the line [tex]\( y = -x \)[/tex].
The correct answer is:
[tex]\[ \boxed{4} \][/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.