Answer :
To find the coordinates of the image [tex]\( D' \)[/tex] after reflecting the point [tex]\( D \)[/tex] over the line [tex]\( y = x \)[/tex], we need to understand the properties of the reflection process.
### Reflection Over the Line [tex]\( y = x \)[/tex]
When a point [tex]\((x, y)\)[/tex] is reflected over the line [tex]\(y = x\)[/tex]:
- The [tex]\( x \)[/tex]-coordinate and [tex]\( y \)[/tex]-coordinate of the original point are swapped.
This means that if you have a point [tex]\(D\)[/tex] with coordinates [tex]\((a, b)\)[/tex], after reflecting it over the line [tex]\(y = x\)[/tex], the coordinates of the image [tex]\(D'\)[/tex] will be [tex]\((b, a)\)[/tex].
Let's confirm this with an example:
#### Example:
- Let the coordinates of point [tex]\(D\)[/tex] be [tex]\((3, 5)\)[/tex].
- After reflecting over the line [tex]\(y = x\)[/tex], the coordinates of [tex]\(D'\)[/tex] would be [tex]\((5, 3)\)[/tex].
So, following the same logic for any general point [tex]\((a, b)\)[/tex]:
### Step-by-Step Solution:
1. Identify the original coordinates of point [tex]\(D\)[/tex]: [tex]\((a, b)\)[/tex].
2. Reflect the point over the line [tex]\(y = x\)[/tex] by swapping the coordinates:
- 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].
Thus, the coordinates of the image [tex]\(D'\)[/tex] after reflection are [tex]\((b, a)\)[/tex].
### Final Answer:
[tex]\( (b, a) \)[/tex]
This matches one of the given answer choices: [tex]\( (b, a) \)[/tex].
### Reflection Over the Line [tex]\( y = x \)[/tex]
When a point [tex]\((x, y)\)[/tex] is reflected over the line [tex]\(y = x\)[/tex]:
- The [tex]\( x \)[/tex]-coordinate and [tex]\( y \)[/tex]-coordinate of the original point are swapped.
This means that if you have a point [tex]\(D\)[/tex] with coordinates [tex]\((a, b)\)[/tex], after reflecting it over the line [tex]\(y = x\)[/tex], the coordinates of the image [tex]\(D'\)[/tex] will be [tex]\((b, a)\)[/tex].
Let's confirm this with an example:
#### Example:
- Let the coordinates of point [tex]\(D\)[/tex] be [tex]\((3, 5)\)[/tex].
- After reflecting over the line [tex]\(y = x\)[/tex], the coordinates of [tex]\(D'\)[/tex] would be [tex]\((5, 3)\)[/tex].
So, following the same logic for any general point [tex]\((a, b)\)[/tex]:
### Step-by-Step Solution:
1. Identify the original coordinates of point [tex]\(D\)[/tex]: [tex]\((a, b)\)[/tex].
2. Reflect the point over the line [tex]\(y = x\)[/tex] by swapping the coordinates:
- 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].
Thus, the coordinates of the image [tex]\(D'\)[/tex] after reflection are [tex]\((b, a)\)[/tex].
### Final Answer:
[tex]\( (b, a) \)[/tex]
This matches one of the given answer choices: [tex]\( (b, a) \)[/tex].
