Final answer:
To map a triangle to a certain position with rigid transformations, translation, rotation, and reflection could all potentially be used, depending on the original positions of the triangle's vertices. Dilation is excluded as it changes the size of the figure, which does not fit the criteria of a rigid transformation.
Step-by-step explanation:
The question relates to identifying possible geometric transformations that could map a triangle to a new position with vertices A at (1, 4) and B at (3, 2). We consider four types of transformations: rotation, translation, dilation, and reflection. Since the question specifies that we are dealing with rigid transformations, dilation can be eliminated because it changes the size of the triangle, whereas rigid transformations preserve distances and angles.
A translation moves every point of a shape the same distance in the same direction. This transformation can change the position of the triangle without altering its shape or size. A rotation turns the triangle around a fixed point, called the center of rotation, again without changing its shape or size. A rotation can be used to move vertex A to (1, 4) and vertex B to (3, 2), if the original positions and the center of rotation are suitable. Lastly, a reflection flips the triangle over a line, known as the line of reflection. Reflection can also position vertices A and B as required, depending on the location of the line of reflection.
Without additional information on the original position of the triangle, it is difficult to determine the precise transformations used. However, since the problem allows for the selection of multiple answers, and all three remaining transformations (other than dilation) could be used to map the original triangle to the new position preserving size and shape, we could consider them as possible solutions.