Certainly! To determine which transformations can be used to carry the point [tex]\( (3,2) \)[/tex] onto itself, let's analyze each transformation one by one.
### A. Translation two units up
- Translation two units up means that every point moves vertically by two units.
- Applying this to the point [tex]\( (3,2) \)[/tex] would move it to [tex]\( (3,4) \)[/tex].
- Since [tex]\( (3,4) \neq (3,2) \)[/tex], this transformation does not carry the point onto itself.
### B. Reflection across the line [tex]\( y = 2 \)[/tex]
- Reflection across the line [tex]\( y = 2 \)[/tex] means that each point [tex]\( (x, y) \)[/tex] is reflected so that its perpendicular distance to the line [tex]\( y = 2 \)[/tex] is the same before and after the reflection.
- For the point [tex]\( (3,2) \)[/tex], it is exactly on the line [tex]\( y = 2 \)[/tex].
- Therefore, reflecting [tex]\( (3,2) \)[/tex] across [tex]\( y = 2 \)[/tex] will leave it unchanged, i.e., it carries [tex]\( (3,2) \)[/tex] onto itself.
### C. Rotation of [tex]\( 180^{\circ} \)[/tex]
- Rotation of [tex]\( 180^{\circ} \)[/tex] around the point [tex]\( (3,2) \)[/tex] means that the point rotates halfway around a circle.
- Since we are rotating around [tex]\( (3,2) \)[/tex] itself, it remains in its own position.
- Therefore, this transformation carries [tex]\( (3,2) \)[/tex] onto itself.
### D. Rotation of [tex]\( 90^{\circ} \)[/tex]
- Rotation of [tex]\( 90^{\circ} \)[/tex] around the point [tex]\( (3,2) \)[/tex] means that each point rotates a quarter turn around it.
- As with the [tex]\( 180^{\circ} \)[/tex] rotation, since we are rotating around [tex]\( (3,2) \)[/tex] itself, it stays at its own position.
- Therefore, this transformation also carries [tex]\( (3,2) \)[/tex] onto itself.
From the analysis:
- Translation two units up does not carry the point onto itself.
- Reflection across the line [tex]\( y = 2 \)[/tex] does carry the point onto itself.
- Rotation of [tex]\( 180^{\circ} \)[/tex] does carry the point onto itself.
- Rotation of [tex]\( 90^{\circ} \)[/tex] does carry the point onto itself.
Thus, the transformations that can be used to carry ABCD onto itself are:
- B. Reflection across the line [tex]\( y = 2 \)[/tex]
- C. Rotation of [tex]\( 180^{\circ} \)[/tex]
- D. Rotation of [tex]\( 90^{\circ} \)[/tex]
Therefore, the final answer is:
[tex]\( \boxed{2, 3, 4} \)[/tex]
