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Which ordered pairs will NOT represent a vertex point in the transformed image *A' B' C* '?

The updated problem involves a triangle with vertices \( A(3, 5) \), \( B(3, 2) \), and \( C(3, 4) \). The triangle is transformed according to the rule \( (x, y) \rightarrow (-y, x) \), which signifies a 90-degree counterclockwise rotation about the origin in the coordinate plane.

Let’s apply this transformation to each of the given vertices:

1. **Transforming vertex \( A(3, 5) \):**
   \[
   A(x, y) = (3, 5) \rightarrow (-y, x) = (-5, 3)
   \]

2. **Transforming vertex \( B(3, 2) \):**
   \[
   B(x, y) = (3, 2) \rightarrow (-y, x) = (-2, 3)
   \]

3. **Transforming vertex \( C(3, 4) \):**
   \[
   C(x, y) = (3, 4) \rightarrow (-y, x) = (-4, 3)
   \]

After the transformation, the new vertices of the triangle are:
- \( A'(-5, 3) \)
- \( B'(-2, 3) \)
- \( C'(-4, 3) \)

The question asks which ordered pairs **do NOT** represent a vertex of the transformed triangle \( A'B'C' \).

The answer choices are:
- \( (-5, 3) \)
- \( (-2, 3) \)
- \( (-4, 3) \)
- \( (-3, 4) \)

Based on the transformation:
- \( (-5, 3) \) is a vertex (corresponding to \( A' \)).
- \( (-2, 3) \) is a vertex (corresponding to \( B' \)).
- \( (-4, 3) \) is a vertex (corresponding to \( C' \)).

However, \( (-3, 4) \) does **NOT** represent any of the transformed vertices. It seems to be a mistaken combination of coordinates from the original points and does not align with the transformation rule.

Thus, the correct answer is:
\[
\boxed{(-3, 4)}
\]

Would you like further clarification on transformations, or is there another aspect you'd like to explore? Here are some related questions:

1. How does rotating a point 180 degrees around the origin affect its coordinates?
2. What would happen if you reflected the triangle over the x-axis instead of rotating it?
3. Could you determine the coordinates of the transformed vertices if a different rule was applied, such as a reflection across the y-axis?
4. How do geometric transformations relate to vector operations?
5. What is the effect of translating (shifting) the triangle after rotating it?

**Tip:** For any transformation, it’s helpful to work through one vertex at a time to ensure each point is accurately transformed before checking which coordinates are valid.

Explore a problem involving rotation of a triangle's vertices in the coordinate plane. Understand how a 90-degree counterclockwise rotation affects coordinates and identify vertices that do not correspond to the transformed shape. Suitable for Grades 9-12.

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