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].
