To solve Sumy's problem of reflecting point [tex]\( D \)[/tex] with coordinates [tex]\( (a, b) \)[/tex] over the line [tex]\( y = x \)[/tex], we need to understand the geometric transformation properties of reflections.
Reflecting a point over the line [tex]\( y = x \)[/tex] swaps its [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates. Here's a detailed step-by-step explanation:
1. Identify the original coordinates of [tex]\( D \)[/tex]:
[tex]\[
D = (a, b)
\][/tex]
2. Reflect the point over the line [tex]\( y = x \)[/tex]:
- When reflecting over the line [tex]\( y = x \)[/tex], the [tex]\( x \)[/tex]-coordinate becomes the [tex]\( y \)[/tex]-coordinate, and the [tex]\( y \)[/tex]-coordinate becomes the [tex]\( x \)[/tex]-coordinate.
- This means the new coordinates, [tex]\( D' \)[/tex], are given by:
[tex]\[
D' = (b, a)
\][/tex]
Thus, after reflecting the point [tex]\( D \)[/tex] over the line [tex]\( y = x \)[/tex], the coordinates of [tex]\( D' \)[/tex] are:
[tex]\[
(b, a)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{(b, a)}
\][/tex]
