Let's determine the rule for reflecting a point across the line [tex]\(y = x\)[/tex].
1. Identify the Original Coordinates:
Suppose we have a point [tex]\(E\)[/tex] with coordinates [tex]\((E_x, E_y)\)[/tex]. For this example, let's consider point [tex]\(E\)[/tex] at [tex]\((-1, 11)\)[/tex].
2. Understanding the Reflection Across [tex]\(y = x\)[/tex]:
When reflecting a point across the line [tex]\(y = x\)[/tex], the [tex]\(x\)[/tex]-coordinate and the [tex]\(y\)[/tex]-coordinate of the point are swapped. Therefore, if the original point is [tex]\((x, y)\)[/tex], its reflection will be [tex]\((y, x)\)[/tex].
3. Applying the Rule to Point [tex]\(E\)[/tex]:
Given the point [tex]\(E\)[/tex], which has coordinates [tex]\((-1, 11)\)[/tex]:
- The [tex]\(x\)[/tex]-coordinate [tex]\((-1)\)[/tex] becomes the new [tex]\(y\)[/tex]-coordinate.
- The [tex]\(y\)[/tex]-coordinate [tex]\(11\)[/tex] becomes the new [tex]\(x\)[/tex]-coordinate.
4. Compute the Reflected Coordinates:
- The new [tex]\(x\)[/tex]-coordinate is [tex]\(11\)[/tex].
- The new [tex]\(y\)[/tex]-coordinate is [tex]\(-1\)[/tex].
So, the coordinates of the reflected point [tex]\(E'\)[/tex] are:
[tex]\[ E' = (11, -1). \][/tex]
Thus, the general rule for reflecting a point [tex]\((x, y)\)[/tex] across the line [tex]\(y = x\)[/tex] is:
[tex]\[ r(x, y) = (y, x). \][/tex]
