Reflecting a point over the line [tex]\( y = x \)[/tex] involves swapping the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates of the point. Let's go through this step-by-step:
1. Understand the reflection over [tex]\( y = x \)[/tex]:
- When reflecting a point [tex]\((x, y)\)[/tex] over the line [tex]\( y = x \)[/tex], the new point becomes [tex]\((y, x)\)[/tex]. This means the [tex]\( x \)[/tex]-coordinate becomes the [tex]\( y \)[/tex]-coordinate and the [tex]\( y \)[/tex]-coordinate becomes the [tex]\( x \)[/tex]-coordinate.
2. Given point [tex]\( D \)[/tex]:
- The coordinates of point [tex]\( D \)[/tex] are [tex]\((a, b)\)[/tex].
3. Swap the coordinates:
- To reflect [tex]\( D \)[/tex] over the line [tex]\( y = x \)[/tex], we swap the [tex]\( x \)[/tex] (which is [tex]\( a \)[/tex]) and [tex]\( y \)[/tex] (which is [tex]\( b \)[/tex]) coordinates. Thus, the coordinates of the reflected point [tex]\( D' \)[/tex] are [tex]\((b, a)\)[/tex].
Therefore, the coordinates of the image [tex]\( D' \)[/tex] after reflecting [tex]\( D \)[/tex] over the line [tex]\( y = x \)[/tex] are [tex]\((b, a)\)[/tex].
So, the correct answer is:
[tex]\( \boxed{(b, a)} \)[/tex]
