To determine the reflection of a point, we need to understand how reflections across different lines affect the coordinates of the point.
Let's analyze the point [tex]\((m, 0)\)[/tex] where [tex]\(m \neq 0\)[/tex], and examine how the coordinates change when reflected across different lines.
1. Reflection across the [tex]\(x\)[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\(x\)[/tex]-axis results in the point [tex]\((x, -y)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex]:
[tex]\[
(m, 0) \rightarrow (m, -0) = (m, 0)
\][/tex]
- The image will be [tex]\((m, 0)\)[/tex], which does not match the required reflected point [tex]\((0, -m)\)[/tex].
2. Reflection across the [tex]\(y\)[/tex]-axis:
- The reflection of a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis results in the point [tex]\((-x, y)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex]:
[tex]\[
(m, 0) \rightarrow (-m, 0)
\][/tex]
- The image will be [tex]\((-m, 0)\)[/tex], which does not match the required reflected point [tex]\((0, -m)\)[/tex].
3. Reflection across the line [tex]\(y = x\)[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] results in the point [tex]\((y, x)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex]:
[tex]\[
(m, 0) \rightarrow (0, m)
\][/tex]
- The image will be [tex]\((0, m)\)[/tex], which does not match the required reflected point [tex]\((0, -m)\)[/tex].
4. Reflection across the line [tex]\(y = -x\)[/tex]:
- The reflection of a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = -x\)[/tex] results in the point [tex]\((-y, -x)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex]:
[tex]\[
(m, 0) \rightarrow (0, -m)
\][/tex]
- The image will be [tex]\((0, -m)\)[/tex], which matches the required reflected point.
Therefore, the reflection of the point [tex]\((m, 0)\)[/tex] across the line [tex]\(y = -x\)[/tex] will produce an image located at [tex]\((0, -m)\)[/tex].
The correct answer is:
- a reflection of the point across the line [tex]\(y = -x\)[/tex].
