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The image of a point is given by the rule [tex]$r_{y=-x}(x, y) \rightarrow(-4, 9)$[/tex]. What are the coordinates of its pre-image?

A. [tex]$(-9, 4)$[/tex]

B. [tex]$(-4, -9)$[/tex]

C. [tex]$(4, 9)$[/tex]

D. [tex]$(9, -4)$[/tex]

Sagot :

GinnyAnsweridn
To find the coordinates of the pre-image of the point under the transformation rule [tex]\(r_{y=-x}\)[/tex], let's examine the given rule.

The rule [tex]\(r_{y=-x}(x, y) \rightarrow (-y, -x)\)[/tex] indicates that under this transformation:
- The x-coordinate becomes the negation of the y-coordinate.
- The y-coordinate becomes the negation of the x-coordinate.

We are given that the point transforms to [tex]\( (-4, 9) \)[/tex].

Now, reverse the transformation to find the pre-image:
- The x-coordinate of the pre-image will be the negated y-coordinate of the image.
- The y-coordinate of the pre-image will be the negated x-coordinate of the image.

Given the image [tex]\( (-4, 9) \)[/tex]:
- The x-coordinate of the pre-image will be [tex]\(-9\)[/tex] (negation of 9).
- The y-coordinate of the pre-image will be [tex]\(4\)[/tex] (negation of -4).

Thus, the coordinates of the pre-image are [tex]\( (-9, 4) \)[/tex].

Therefore, the correct answer is:
[tex]\[ \boxed{(-9, 4)} \][/tex]
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