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Sumy is working in geometry class and is given figure ABCD in the coordinate plane to reflect. The coordinates of point [tex]\( D \)[/tex] are [tex]\( (a, b) \)[/tex]. 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]



Answer :

GinnyAnswer
To determine the coordinates of the image [tex]\( D' \)[/tex] after reflecting point [tex]\( D \)[/tex] over the line [tex]\( y = x \)[/tex], we need to follow specific steps involved in the reflection process.

When a point [tex]\((x, y)\)[/tex] is reflected over the line [tex]\( y = x \)[/tex], the coordinates of the point are swapped. This means that the [tex]\( x \)[/tex]-coordinate becomes the [tex]\( y \)[/tex]-coordinate and the [tex]\( y \)[/tex]-coordinate becomes the [tex]\( x \)[/tex]-coordinate.

Let’s apply this rule to point [tex]\( D \)[/tex] whose coordinates are [tex]\((a, b)\)[/tex].

1. Initial Coordinates: Point [tex]\( D \)[/tex] has coordinates [tex]\((a, b)\)[/tex].
2. Reflection Process: Swap the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-coordinates of point [tex]\( D \)[/tex]:
- The new [tex]\( x \)[/tex]-coordinate will be the original [tex]\( y \)[/tex]-coordinate [tex]\( b \)[/tex].
- The new [tex]\( y \)[/tex]-coordinate will be the original [tex]\( x \)[/tex]-coordinate [tex]\( a \)[/tex].

Therefore, the coordinates of the reflected point [tex]\( D' \)[/tex] are [tex]\((b, a)\)[/tex].

To match the options given:
- [tex]\((a, -b)\)[/tex]
- [tex]\((b, a)\)[/tex]
- [tex]\((-a, b)\)[/tex]
- [tex]\((-b, -a)\)[/tex]

The correct coordinates of point [tex]\( D' \)[/tex] after reflection are [tex]\((b, a)\)[/tex].

Thus, the coordinates of the image [tex]\( D' \)[/tex] are [tex]\(\boxed{(b, a)}\)[/tex].

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