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.