To determine which transformation or sequence of transformations maps triangle $\triangle ABC$ to $\triangle A'B'C'$, let’s analyze the movement of each vertex:

1. Initial Observation of Points:

   • $A(5, 5) \rightarrow A'(0, -5)$
   • $B(1, 3) \rightarrow B'(-4, -3)$
   • $C(2, 1) \rightarrow C'(-3, -1)$

2. Testing Possible Transformations:

   • Option A (a translation 5 units left) would move each point 5 units left on the $x$-axis. However, this would not place the points correctly, as the $y$-coordinates of the points in $\triangle A'B'C'$ differ significantly from those in $\triangle ABC$.
   • Option B (a reflection across $x = 0$, then a translation 5 units left) would reflect the triangle across the $y$-axis (changing the sign of the $x$-coordinates), and then translating left. After this reflection, $A$, $B$, and $C$ would still not be correctly aligned with $A'B'C'$.
   • Option C (a reflection across $y = 0$, then a translation 5 units left):
     • Reflecting across $y = 0$ (the $x$-axis) changes the sign of the $y$-coordinates.
       • $A(5, 5) \rightarrow (5, -5)$
       • $B(1, 3) \rightarrow (1, -3)$
       • $C(2, 1) \rightarrow (2, -1)$
     • Then, translating each point 5 units left results in:
       • $(5, -5) \rightarrow (0, -5)$ (matching $A'$)
       • $(1, -3) \rightarrow (-4, -3)$ (matching $B'$)
       • $(2, -1) \rightarrow (-3, -1)$ (matching $C'$)
     • This transformation sequence maps $\triangle ABC$ to $\triangle A'B'C'$ exactly.
   • Option D (a rotation 90° about the origin) would not yield the correct coordinates for $A'B'C'$.

3. Conclusion:

   • The correct answer is C: a reflection across $y = 0$, then a translation 5 units left.

Would you like more details on the steps, or have any questions?

Related Questions

1. How do reflections affect the coordinates of points in the coordinate plane?
2. What is the effect of a translation on the coordinates of a shape?
3. How would a 90° rotation about the origin change the coordinates of a point?
4. Can a transformation sequence include more than one type of transformation?
5. How do you determine which sequence of transformations is correct?

Tip:

When dealing with transformations, it's helpful to break down each transformation step-by-step and verify how it affects each coordinate of the points involved.