To find the reflection of the point [tex]\( P = (-2, 7) \)[/tex] across the line [tex]\( x = 1 \)[/tex], we follow these steps:
1. Identify the given point and the line of reflection:
- The given point is [tex]\( P = (-2, 7) \)[/tex].
- The line of reflection is [tex]\( x = 1 \)[/tex].
2. Calculate the distance of the point from the line of reflection:
- The [tex]\( x \)[/tex]-coordinate of the point [tex]\( P \)[/tex] is [tex]\(-2\)[/tex].
- The line [tex]\( x = 1 \)[/tex] tells us that all points on this line have an [tex]\( x \)[/tex]-coordinate of 1.
- The distance from [tex]\(-2\)[/tex] to 1 is calculated as [tex]\( 1 - (-2) = 1 + 2 = 3 \)[/tex].
3. Reflect the [tex]\( x \)[/tex]-coordinate:
- To reflect the point across the line [tex]\( x = 1 \)[/tex], we need to move the point the same distance to the other side of the line.
- Since the distance from [tex]\(-2\)[/tex] to 1 is 3 units, we move 3 units to the right of 1.
- The new [tex]\( x \)[/tex]-coordinate is [tex]\( 1 + 3 = 4 \)[/tex].
4. Retain the [tex]\( y \)[/tex]-coordinate:
- The reflection process does not change the [tex]\( y \)[/tex]-coordinate.
- Therefore, the [tex]\( y \)[/tex]-coordinate remains 7.
5. Obtain the reflected point:
- The coordinates of the reflected point are [tex]\( (4, 7) \)[/tex].
So, [tex]\( R_{x=1}(P) = (4, 7) \)[/tex].
Thus, the reflected point in coordinate form is [tex]\( ([4], [7]) \)[/tex].
