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Sumy is working in geometry class and is given figure [tex]$ABCD$[/tex] in the coordinate plane to reflect. The coordinates of point [tex]$D$[/tex] are [tex]$(a, b)$[/tex] and she reflects the figure over the line [tex]$y=x$[/tex]. What are the coordinates of the image [tex]$D'$[/tex]?

A. [tex]$(a, -b)$[/tex]
B. [tex]$(b, a)$[/tex]
C. [tex]$(-a, b)$[/tex]
D. [tex]$(-b, -a)$[/tex]

Sagot :

GinnyAnswer
When reflecting a point [tex]\((a, b)\)[/tex] over the line [tex]\(y = x\)[/tex], you swap the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates of the point. This means the [tex]\(x\)[/tex]-coordinate of the image will be [tex]\(b\)[/tex] and the [tex]\(y\)[/tex]-coordinate of the image will be [tex]\(a\)[/tex].

Given the coordinates of point [tex]\(D\)[/tex] as [tex]\((a, b)\)[/tex], the steps to find the coordinates of the reflected image [tex]\(D'\)[/tex] are as follows:

1. Start with the coordinates of the original point [tex]\(D\)[/tex], which are [tex]\((a, b)\)[/tex].
2. Reflect the point over the line [tex]\(y = x\)[/tex]; to do this, you swap the [tex]\(x\)[/tex]-coordinate and [tex]\(y\)[/tex]-coordinate.
3. After swapping, the new coordinates are [tex]\((b, a)\)[/tex].

Thus, the coordinates of the image point [tex]\(D'\)[/tex] after reflecting [tex]\(D(a, b)\)[/tex] over the line [tex]\(y = x\)[/tex] are [tex]\((b, a)\)[/tex].

Therefore, the correct answer is [tex]\((b, a)\)[/tex].
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