Exercise 3:

To map $\triangle ABC$ to $\triangle A'B'C'$:

1. Translation: Move each point 8 units to the right and 4 units up. This shifts:
   • $A(-6, 4)$ to $A'(2, 8)$,
   • $B(-8, -2)$ to $B'(0, 2)$,
   • $C(-4, -2)$ to $C'(4, 2)$.

Thus, the transformation is a translation by $(+8, +4)$.

Exercise 4:

To map $\triangle ABC$ to $\triangle A'B'C'$:

1. Reflection: Reflect $\triangle ABC$ over the $x$-axis. This transformation maps:
   • $A(4, 8)$ to $A'(4, -8)$,
   • $B(-4, 8)$ to $B'(-4, -8)$,
   • $C(4, 4)$ to $C'(4, -4)$.

Thus, the transformation is a reflection over the $x$-axis.

Do you want a detailed breakdown of these transformations or have further questions?

Questions to Expand Understanding:

1. What is the general formula for a translation in the coordinate plane?
2. How do reflections change the coordinates of points relative to axes?
3. Can a sequence of transformations ever result in a non-congruent image? Why or why not?
4. How can you verify that two triangles are congruent after a transformation?
5. Are there cases where a reflection and translation combined yield the same result?

Tip:

Always compare corresponding points in the preimage and image to determine the exact transformation.