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Complete the mapping of the vertices.

What is the rule that describes a reflection over the line [tex]\( y = x \)[/tex]?

[tex]\( r(x, y) = \square \)[/tex]

Sagot :

GinnyAnswer
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]
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