Answer :
To determine which transformations can be used to carry the quadrilateral [tex]\(ABCD\)[/tex] onto itself with the point of rotation being [tex]\((3, 2)\)[/tex], let's evaluate each option one by one and consider the impact on the figure:
### Option A: Translation two units down
Translation involves shifting every point of the figure by a certain distance in a specific direction. Translating [tex]\(ABCD\)[/tex] two units down means shifting every point down by two units along the [tex]\(y\)[/tex]-axis. This action will move the entire figure such that it no longer coincides with its original position. Therefore, this transformation will not map [tex]\(ABCD\)[/tex] onto itself.
### Option B: Reflection across the line [tex]\(x = 3\)[/tex]
Reflecting across the line [tex]\(x = 3\)[/tex] means each point of the figure is mirrored across the vertical line [tex]\(x=3\)[/tex]. This line acts as an axis of symmetry for the quadrilateral [tex]\(ABCD\)[/tex]. When reflected over this line, each point on one side of the line will move to a corresponding point on the opposite side at the same distance from the line. This reflection maintains the shape and relative positioning of [tex]\(ABCD\)[/tex], effectively mapping the figure onto itself. Therefore, this transformation can be used to carry [tex]\(ABCD\)[/tex] onto itself.
### Option C: Rotation of [tex]\(90^\circ\)[/tex]
A rotation of [tex]\(90^\circ\)[/tex] around the point [tex]\((3, 2)\)[/tex] will pivot the entire figure by 90 degrees either clockwise or counterclockwise. Given the specificity of the motion, this changes the orientation of the figure such that the vertices are repositioned to new coordinates, which generally does not coincide with the figure’s original outline. Thus, a [tex]\(90^\circ\)[/tex] rotation will not map [tex]\(ABCD\)[/tex] onto itself.
### Option D: Reflection across the line [tex]\(y = 2\)[/tex]
Reflecting across the line [tex]\(y = 2\)[/tex] means each point of the figure is mirrored across the horizontal line [tex]\(y=2\)[/tex]. Similar to the reflection over [tex]\(x=3\)[/tex], this line also acts as an axis of symmetry. The symmetry here ensures that points above the line move to a corresponding point below the line and vice-versa, thus preserving the figure’s structure and relative positioning. Therefore, this reflection indeed maps [tex]\(ABCD\)[/tex] onto itself.
### Conclusion
Based on the analysis:
- Option A (Translation two units down) will not map [tex]\(ABCD\)[/tex] onto itself.
- Option B (Reflection across the line [tex]\(x=3\)[/tex]) will map [tex]\(ABCD\)[/tex] onto itself.
- Option C (Rotation of [tex]\(90^\circ\)[/tex]) will not map [tex]\(ABCD\)[/tex] onto itself.
- Option D (Reflection across the line [tex]\(y=2\)[/tex]) will map [tex]\(ABCD\)[/tex] onto itself.
Thus, the transformations that can be used to carry [tex]\(ABCD\)[/tex] onto itself are:
[tex]\[ \boxed{B \text{ and } D} \][/tex]
### Option A: Translation two units down
Translation involves shifting every point of the figure by a certain distance in a specific direction. Translating [tex]\(ABCD\)[/tex] two units down means shifting every point down by two units along the [tex]\(y\)[/tex]-axis. This action will move the entire figure such that it no longer coincides with its original position. Therefore, this transformation will not map [tex]\(ABCD\)[/tex] onto itself.
### Option B: Reflection across the line [tex]\(x = 3\)[/tex]
Reflecting across the line [tex]\(x = 3\)[/tex] means each point of the figure is mirrored across the vertical line [tex]\(x=3\)[/tex]. This line acts as an axis of symmetry for the quadrilateral [tex]\(ABCD\)[/tex]. When reflected over this line, each point on one side of the line will move to a corresponding point on the opposite side at the same distance from the line. This reflection maintains the shape and relative positioning of [tex]\(ABCD\)[/tex], effectively mapping the figure onto itself. Therefore, this transformation can be used to carry [tex]\(ABCD\)[/tex] onto itself.
### Option C: Rotation of [tex]\(90^\circ\)[/tex]
A rotation of [tex]\(90^\circ\)[/tex] around the point [tex]\((3, 2)\)[/tex] will pivot the entire figure by 90 degrees either clockwise or counterclockwise. Given the specificity of the motion, this changes the orientation of the figure such that the vertices are repositioned to new coordinates, which generally does not coincide with the figure’s original outline. Thus, a [tex]\(90^\circ\)[/tex] rotation will not map [tex]\(ABCD\)[/tex] onto itself.
### Option D: Reflection across the line [tex]\(y = 2\)[/tex]
Reflecting across the line [tex]\(y = 2\)[/tex] means each point of the figure is mirrored across the horizontal line [tex]\(y=2\)[/tex]. Similar to the reflection over [tex]\(x=3\)[/tex], this line also acts as an axis of symmetry. The symmetry here ensures that points above the line move to a corresponding point below the line and vice-versa, thus preserving the figure’s structure and relative positioning. Therefore, this reflection indeed maps [tex]\(ABCD\)[/tex] onto itself.
### Conclusion
Based on the analysis:
- Option A (Translation two units down) will not map [tex]\(ABCD\)[/tex] onto itself.
- Option B (Reflection across the line [tex]\(x=3\)[/tex]) will map [tex]\(ABCD\)[/tex] onto itself.
- Option C (Rotation of [tex]\(90^\circ\)[/tex]) will not map [tex]\(ABCD\)[/tex] onto itself.
- Option D (Reflection across the line [tex]\(y=2\)[/tex]) will map [tex]\(ABCD\)[/tex] onto itself.
Thus, the transformations that can be used to carry [tex]\(ABCD\)[/tex] onto itself are:
[tex]\[ \boxed{B \text{ and } D} \][/tex]
