To determine the coordinates of the image [tex]\(D'\)[/tex] when point [tex]\(D(a, b)\)[/tex] is reflected over the line [tex]\(y = x\)[/tex], let's go through the steps involved in such a reflection.
1. Understanding reflection over the line [tex]\(y = x\)[/tex]:
- In general, when a point [tex]\((x, y)\)[/tex] is reflected over the line [tex]\(y = x\)[/tex], its new coordinates become [tex]\((y, x)\)[/tex].
- This means that the [tex]\(x\)[/tex]- and [tex]\(y\)[/tex]-coordinates of the point are swapped.
2. Applying the reflection rule:
- Given a point [tex]\(D\)[/tex] with coordinates [tex]\((a, b)\)[/tex]:
- The [tex]\(a\)[/tex]-coordinate of [tex]\(D\)[/tex] corresponds to the [tex]\(x\)[/tex]-coordinate.
- The [tex]\(b\)[/tex]-coordinate of [tex]\(D\)[/tex] corresponds to the [tex]\(y\)[/tex]-coordinate.
3. Finding the new coordinates:
- When [tex]\(D(a, b)\)[/tex] is reflected over the line [tex]\(y = x\)[/tex]:
- The [tex]\(a\)[/tex] (or [tex]\(x\)[/tex]) coordinate moves to the [tex]\(y\)[/tex] position.
- The [tex]\(b\)[/tex] (or [tex]\(y\)[/tex]) coordinate moves to the [tex]\(x\)[/tex] position.
- Therefore, the new coordinates for [tex]\(D'\)[/tex] are obtained by swapping [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
4. Result:
- The new coordinates of [tex]\(D'\)[/tex] will be [tex]\((b, a)\)[/tex].
Thus, after reflecting the point [tex]\(D(a, b)\)[/tex] over the line [tex]\(y = x\)[/tex], the coordinates of the image [tex]\(D'\)[/tex] are [tex]\((b, a)\)[/tex].
The correct answer is:
[tex]\[ \boxed{(b, a)} \][/tex]
