Final answer:
All of the listed transformations - reflection, translation, and rotation - can carry a triangle onto itself if they're applied correctly, not altering the shape or size of the triangle.
Step-by-step explanation:
The set of transformations that will carry a triangle onto itself depends on the specific properties of the triangle and the type of transformations applied. Let's explore each option:
a) Reflection over a line:
- A triangle can be reflected over a line, and this transformation can carry the triangle onto itself if the line of reflection passes through the centroid or an incenter of the triangle.
b) Translation:
- A translation involves moving the entire triangle along a straight path without any rotation or reflection. If the translation vector is such that the triangle lands back on itself, then a translation can carry the triangle onto itself.
c) Rotation:
- A rotation about the centroid or an incenter of the triangle could carry the triangle onto itself.
d) All of the above:
- If a combination of reflection, translation, and rotation is applied in a specific way, it is possible for the triangle to be carried onto itself.
In summary, the correct answer is:
d) All of the above
The specific transformations that will achieve this depend on the characteristics of the triangle and the chosen transformation parameters.