Answer :
To determine which transformations can carry the quadrilateral [tex]\(ABCD\)[/tex] onto itself, we'll analyze each given transformation option and verify whether it can map the quadrilateral back onto itself.
### Option A: Translation four units to the right
A translation four units to the right involves shifting every point of the figure horizontally by four units. Translation moves the entire figure to a new location without changing its orientation or size. Since the figure is relocated, it will not map back onto its original position. Therefore, translation four units to the right is not a valid transformation.
### Option B: Reflection across the line [tex]\(x=3\)[/tex]
Reflecting a figure across the line [tex]\(x=3\)[/tex] involves creating a mirror image of the figure with respect to this vertical line located at [tex]\(x=3\)[/tex]. If the quadrilateral [tex]\(ABCD\)[/tex] is symmetric with respect to this line, the reflection will map the figure onto itself. Symmetry about the line [tex]\(x=3\)[/tex] would mean that each point on one side of the line has a corresponding point on the opposite side at the same distance from the line. Hence, reflection across the line [tex]\(x=3\)[/tex] is a valid transformation.
### Option C: Dilation by a factor of 2
Dilation changes the size of the figure while maintaining its shape. In this case, dilating [tex]\(ABCD\)[/tex] by a factor of 2 will enlarge the distances between all points of the figure by twice, making the figure larger overall but not preserving its original size and position. Consequently, dilation cannot map the shape back onto itself while maintaining its size. Therefore, dilation by a factor of 2 is not a valid transformation.
### Option D: Rotation of [tex]\(180^{\circ}\)[/tex]
Rotating the quadrilateral [tex]\(ABCD\)[/tex] by [tex]\(180^{\circ}\)[/tex] around the point [tex]\((3,2)\)[/tex] involves turning the figure around this central point so that every point moves along a circular arc of [tex]\(180^{\circ}\)[/tex] and ends up in the diametrically opposite position. If the quadrilateral is symmetric with respect to a [tex]\(180^{\circ}\)[/tex] rotation around this point, then the rotation will map the figure onto itself. Therefore, rotation of [tex]\(180^{\circ}\)[/tex] about the point [tex]\((3,2)\)[/tex] is a valid transformation.
### Conclusion
Based on our analysis, the valid transformations that can carry [tex]\(ABCD\)[/tex] onto itself are:
- B. Reflection across the line [tex]\(x=3\)[/tex]
- D. Rotation of [tex]\(180^{\circ}\)[/tex]
There are exactly two such transformations. Thus, the total number of valid transformations is:
[tex]\[ \boxed{2} \][/tex]
### Option A: Translation four units to the right
A translation four units to the right involves shifting every point of the figure horizontally by four units. Translation moves the entire figure to a new location without changing its orientation or size. Since the figure is relocated, it will not map back onto its original position. Therefore, translation four units to the right is not a valid transformation.
### Option B: Reflection across the line [tex]\(x=3\)[/tex]
Reflecting a figure across the line [tex]\(x=3\)[/tex] involves creating a mirror image of the figure with respect to this vertical line located at [tex]\(x=3\)[/tex]. If the quadrilateral [tex]\(ABCD\)[/tex] is symmetric with respect to this line, the reflection will map the figure onto itself. Symmetry about the line [tex]\(x=3\)[/tex] would mean that each point on one side of the line has a corresponding point on the opposite side at the same distance from the line. Hence, reflection across the line [tex]\(x=3\)[/tex] is a valid transformation.
### Option C: Dilation by a factor of 2
Dilation changes the size of the figure while maintaining its shape. In this case, dilating [tex]\(ABCD\)[/tex] by a factor of 2 will enlarge the distances between all points of the figure by twice, making the figure larger overall but not preserving its original size and position. Consequently, dilation cannot map the shape back onto itself while maintaining its size. Therefore, dilation by a factor of 2 is not a valid transformation.
### Option D: Rotation of [tex]\(180^{\circ}\)[/tex]
Rotating the quadrilateral [tex]\(ABCD\)[/tex] by [tex]\(180^{\circ}\)[/tex] around the point [tex]\((3,2)\)[/tex] involves turning the figure around this central point so that every point moves along a circular arc of [tex]\(180^{\circ}\)[/tex] and ends up in the diametrically opposite position. If the quadrilateral is symmetric with respect to a [tex]\(180^{\circ}\)[/tex] rotation around this point, then the rotation will map the figure onto itself. Therefore, rotation of [tex]\(180^{\circ}\)[/tex] about the point [tex]\((3,2)\)[/tex] is a valid transformation.
### Conclusion
Based on our analysis, the valid transformations that can carry [tex]\(ABCD\)[/tex] onto itself are:
- B. Reflection across the line [tex]\(x=3\)[/tex]
- D. Rotation of [tex]\(180^{\circ}\)[/tex]
There are exactly two such transformations. Thus, the total number of valid transformations is:
[tex]\[ \boxed{2} \][/tex]
