To find the coordinates of the image of point \(D\) after it has been reflected over the line \(y = x\), we need to understand the geometric transformation that occurs during this reflection.
1. Understanding Reflection Over \(y = x\):
When a point \((x, y)\) is reflected over the line \(y = x\), it gets mapped to a new point where the \(x\) and \(y\) coordinates are swapped. In essence, the coordinates \((x, y)\) become \((y, x)\).
2. Applying This Transformation:
Given the point \(D\) with coordinates \((a, b)\):
- The \(x\)-coordinate \(a\) will become the \(y\)-coordinate of the reflected point.
- The \(y\)-coordinate \(b\) will become the \(x\)-coordinate of the reflected point.
Therefore, when we reflect \((a, b)\) over the line \(y = x\), the new coordinates of the point \(D^{\prime}\) will be \((b, a)\).
3. Conclusion:
The coordinates of the image \(D^{\prime}\) after reflecting point \(D\) \((a, b)\) over the line \(y = x\) are \((b, a)\).
Thus, the correct answer is:
[tex]$[tex]$\boxed{(b, a)}$[/tex]$[/tex]
