To determine the line of reflection for point [tex]\( P = (4, -5) \)[/tex] and its reflection point [tex]\( P' = (-4, -5) \)[/tex], follow these steps:
1. Identify the Coordinates of [tex]\( P \)[/tex] and [tex]\( P' \)[/tex]:
- [tex]\( P = (4, -5) \)[/tex]
- [tex]\( P' = (-4, -5) \)[/tex]
2. Analyze the Transformation:
- The x-coordinate of [tex]\( P \)[/tex] is 4, and for [tex]\( P' \)[/tex], it is -4.
- The y-coordinate remains the same for both points, which is -5.
3. Determine Reflection Properties:
- When a point is reflected across the y-axis, the x-coordinate changes sign, while the y-coordinate remains unchanged.
4. Match with Possible Lines of Reflection:
- If the point was reflected across the y-axis:
- The x-coordinate changes sign: (4 becomes -4)
- The y-coordinate remains the same: (-5 stays -5)
- This transformation matches our points [tex]\( P \)[/tex] and [tex]\( P' \)[/tex].
- If the point was reflected across the x-axis:
- The y-coordinate would change sign.
- The x-coordinate remains the same.
- This does not match our transformation.
- If the point was reflected across the line [tex]\( y = x \)[/tex] or [tex]\( y = -x \)[/tex]:
- Both the x and y coordinates would interchange or change sign and swap, which is not applicable here.
5. Conclusion:
Based on the analysis, the line of reflection that correctly transforms point [tex]\( P \)[/tex] to [tex]\( P' \)[/tex] is the y-axis.
Thus, the line of reflection for [tex]\( P = (4, -5) \)[/tex] and [tex]\( P' = (-4, -5) \)[/tex] is the:
y-axis
