Use the coordinates to answer the question.

Question

Triangle ABC:
Point A: (0, 5)
Point B: (5, 7)
Point C: (7, 0)

Triangle A'B'C':
Point A': (0, 2.5)
Point B': (2.5, 3.5)
Point C': (3.5, 0)

What type of transformation can map triangle ABC -> triangle A'B'C'?
A. a rotation
B. no rigid transformation can make this transformation
C. a translation
D. a reflection

Answer 1

To determine what type of transformation maps triangle ABC to triangle A'B'C', we need to analyze the coordinates of the points in both triangles.

First, we will look at the coordinates of both triangles:

• Triangle ABC:

  • A: (0, 5)
  • B: (5, 7)
  • C: (7, 0)

• Triangle A'B'C':

  • A': (0, 2.5)
  • B': (2.5, 3.5)
  • C': (3.5, 0)

Next, we can observe that:

1. The y-coordinates of points A and A' remain the same (0, 5) -> (0, 2.5), but the y-coordinate of A' is halved.
2. The points B and B' are not congruent since (5, 7) and (2.5, 3.5) do not have the same distance from the origin or proportions as seen from the original triangle.
3. The same goes for point C to C' where (7, 0) -> (3.5, 0); point C' is also halved in its x-coordinate.

Observing these changes, we can conclude that triangle A'B'C' is similar but smaller than triangle ABC, suggesting that there has been a dilation (scaling transformation), which is not among the given options.

Since no rigid transformation (rotation, reflection, or translation) can achieve the halving of all x and y coordinates of the points, the correct answer is:

B. no rigid transformation can make this transformation.