Let's go through the different parts of the problem based on the image provided:

1. Triangles ABC and A'B'C'

(a) Coordinates of A, B, and C

• From the diagram, we can observe the coordinates:
  • $A = (-2, 2)$
  • $B = (1, 1)$
  • $C = (0, 5)$

(b) Translation to A'B'C'

The triangle $ABC$ is translated 7 units to the right and 6 units down.

• To translate, you adjust the x-coordinates by adding 7 and the y-coordinates by subtracting 6.
  • $A' = (-2 + 7, 2 - 6) = (5, -4)$
  • $B' = (1 + 7, 1 - 6) = (8, -5)$
  • $C' = (0 + 7, 5 - 6) = (7, -1)$

(c) Coordinates of A', B', and C'

Using the translation:

• $A' = (5, -4)$
• $B' = (8, -5)$
• $C' = (7, -1)$

(d) Are triangles $ABC$ and $A'B'C'$ congruent?

• Since the translation preserves shape and size, the triangles $ABC$ and $A'B'C'$ are congruent.

(e) Is $\triangle ABC \parallel \triangle A'B'C'$?

• Yes, the triangles are parallel since the transformation is a translation, which keeps corresponding sides parallel.

2. Points L, M, and N

(a) Coordinates of L, M, and N

From the second diagram:

• $L = (5, 4)$
• $M = (3, -2)$
• $N = (-3, -4)$

(b) Reflection across the x-axis

The reflection of a point across the x-axis changes the sign of its y-coordinate while keeping the x-coordinate the same.

• $L' = (5, -4)$
• $M' = (3, 2)$
• $N' = (-3, 4)$

(c) Complete the table

(d) Relationship between the coordinates after reflection

• The reflection of a point across the x-axis results in the y-coordinate changing its sign, while the x-coordinate remains the same.

(e) General formula for reflection in the x-axis

• If $P(x, y)$ is reflected across the x-axis, the coordinates of the reflected point $P'$ are $(x, -y)$.

Let me know if you want to go deeper into any part or have more questions. Here are 5 relative questions to explore:

1. How would the coordinates change if triangle $ABC$ was rotated instead of translated?
2. What is the general rule for reflecting a point across the y-axis?
3. Can you reflect a shape across both axes? How would that affect the coordinates?
4. Are translations the only transformations that preserve the congruence of shapes?
5. What is the difference between translations and rotations in terms of coordinate changes?

Tip: Always visualize transformations on a graph to clearly see how points and shapes shift or reflect!