Answered

IDNLearn.com provides a user-friendly platform for finding and sharing knowledge. Get the information you need quickly and accurately with our reliable and thorough Q&A platform.

A point has the coordinates [tex]\((m, 0)\)[/tex] and [tex]\(m \neq 0\)[/tex]. Which reflection of the point will produce an image located at [tex]\((0, -m)\)[/tex]?

A. A reflection of the point across the [tex]\(x\)[/tex]-axis
B. A reflection of the point across the [tex]\(y\)[/tex]-axis
C. A reflection of the point across the line [tex]\(y=x\)[/tex]
D. A reflection of the point across the line [tex]\(y=-x\)[/tex]

Sagot :

GinnyAnsweridn
To determine which reflection of the point [tex]\((m, 0)\)[/tex] will produce an image located at [tex]\((0, -m)\)[/tex], we need to consider the effect of reflecting a point across different lines.

1. Reflection across the [tex]\(x\)[/tex]-axis:
- Reflecting a point [tex]\((a, b)\)[/tex] across the [tex]\(x\)[/tex]-axis changes its coordinates to [tex]\((a, -b)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], the reflected coordinates would be [tex]\((m, -0) = (m, 0)\)[/tex].
- Clearly, this reflection does not change the point, so it does not produce the desired image [tex]\((0, -m)\)[/tex].

2. Reflection across the [tex]\(y\)[/tex]-axis:
- Reflecting a point [tex]\((a, b)\)[/tex] across the [tex]\(y\)[/tex]-axis changes its coordinates to [tex]\((-a, b)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], the reflected coordinates would be [tex]\((-m, 0)\)[/tex].
- This does not match the desired image coordinates of [tex]\((0, -m)\)[/tex].

3. Reflection across the line [tex]\(y = x\)[/tex]:
- Reflecting a point [tex]\((a, b)\)[/tex] across the line [tex]\(y = x\)[/tex] swaps the coordinates to [tex]\((b, a)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], the reflected coordinates would be [tex]\((0, m)\)[/tex].
- Again, this does not match the desired image coordinates of [tex]\((0, -m)\)[/tex].

4. Reflection across the line [tex]\(y = -x\)[/tex]:
- Reflecting a point [tex]\((a, b)\)[/tex] across the line [tex]\(y = -x\)[/tex] swaps and negates the coordinates to [tex]\((-b, -a)\)[/tex].
- For the point [tex]\((m, 0)\)[/tex], the reflected coordinates would be [tex]\((0, -m)\)[/tex].
- This matches the desired image coordinates of [tex]\((0, -m)\)[/tex].

Hence, the correct reflection that produces the image at [tex]\((0, -m)\)[/tex] is a reflection of the point across the line [tex]\(y = -x\)[/tex]. Therefore, the answer is:

a reflection of the point across the line [tex]\(y = -x\)[/tex]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Thank you for choosing IDNLearn.com. We’re committed to providing accurate answers, so visit us again soon.