To reflect a point across the line [tex]\(x-y=0\)[/tex], it's helpful to recognize that this line can be rewritten in a more familiar form, which is [tex]\(y=x\)[/tex]. Reflecting a point across [tex]\(y=x\)[/tex] involves interchanging the coordinates of the point.
Given point [tex]\(E(4, 5)\)[/tex]:
1. Identify the coordinates of the point [tex]\(E\)[/tex]. The point [tex]\(E\)[/tex] has coordinates [tex]\((4, 5)\)[/tex].
2. To reflect [tex]\(E\)[/tex] across the line [tex]\(y=x\)[/tex], we need to swap the x-coordinate and the y-coordinate of the point. This means that:
- The x-coordinate of [tex]\(E\)[/tex] is 4.
- The y-coordinate of [tex]\(E\)[/tex] is 5.
When reflected across the line [tex]\(y=x\)[/tex], these coordinates will be interchanged.
3. Therefore, after reflection:
- The new x-coordinate will be the original y-coordinate of [tex]\(E\)[/tex], which is 5.
- The new y-coordinate will be the original x-coordinate of [tex]\(E\)[/tex], which is 4.
Thus, the coordinates of the reflected point [tex]\(E'\)[/tex] are [tex]\((5, 4)\)[/tex].
Therefore, the point [tex]\(E(4, 5)\)[/tex] reflected across the line [tex]\(x-y=0\)[/tex] (or [tex]\(y=x\)[/tex]) is [tex]\((5, 4)\)[/tex].
