To solve this problem, let's determine the translation vector from point $P(2, 3)$ to point $P'(5, -4)$, and then apply the same translation to point $(6, -2)$.

1. Find the Translation Vector:

   • The change in the $x$-coordinates from $P$ to $P'$ is $5 - 2 = 3$.
   • The change in the $y$-coordinates from $P$ to $P'$ is $-4 - 3 = -7$.
   • Therefore, the translation vector is $(3, -7)$.

2. Apply the Translation Vector to (6, -2):

   • New $x$-coordinate: $6 + 3 = 9$.
   • New $y$-coordinate: $-2 - 7 = -9$.

So, the image of $(6, -2)$ under the same translation is $(9, -9)$.

Answer:

The correct option is:

• C. $(9, -9)$

Would you like a further breakdown of the solution or have any questions?

Further Questions

1. What is the effect of different translation vectors on coordinates?
2. How would the solution change if the coordinates of the points were different?
3. How can translations be visualized on a coordinate plane?
4. How do translations differ from other transformations like rotations or reflections?
5. Can translations affect the size and shape of figures?

Tip

Remember, in translations, every point in the shape or figure moves by the same vector, preserving size and shape.