In this problem, we need to classify each transformation rule of triangle $\triangle ABC$ to determine if it results in a congruent or non-congruent triangle $\triangle A'B'C'$.

Congruent Transformations:

Congruent transformations are those that preserve the size and shape of the figure, such as:

1. Translations (shifting the figure without changing its size or orientation).
2. Rotations (turning the figure around a point).
3. Reflections (flipping the figure over a line).

Non-Congruent Transformations:

Non-congruent transformations change the size of the figure, which means the triangles will not be congruent. These include:

1. Dilations (changing the size of the figure but preserving its shape).
2. Other non-isometric transformations (such as compressions or expansions).

Let's classify each rule:

1. $(x, y) \to (x + 5, y)$ – This is a translation, shifting the figure to the right by 5 units. Congruent.
2. $(x, y) \to (5x, 5y)$ – This is a dilation (scaling the figure by a factor of 5). Not congruent.
3. $(x, y) \to (0.5x, 0.5y)$ – This is a dilation (scaling down by a factor of 0.5). Not congruent.
4. $(x, y) \to (x, -y)$ – This is a reflection across the x-axis. Congruent.
5. $(x, y) \to (-x, -y)$ – This is a rotation by 180° around the origin. Congruent.

Final Classification:

• Congruent:

  • $(x, y) \to (x + 5, y)$
  • $(x, y) \to (x, -y)$
  • $(x, y) \to (-x, -y)$

• Not Congruent:

  • $(x, y) \to (5x, 5y)$
  • $(x, y) \to (0.5x, 0.5y)$

Would you like further details or have any questions about the classification?

Related Questions:

1. What are other examples of congruent transformations?
2. How does a reflection preserve congruence in a figure?
3. How do you calculate the scale factor for dilations?
4. Can a dilation result in a congruent figure if the scale factor is 1?
5. What happens to angles and sides under a rotation?

Tip:

When identifying congruence, remember that transformations preserving distances between points result in congruent figures.