From the image, here is a breakdown of the problem and solution provided:

Part 1 of 3 (Translation Coordinates)

The task here was to fill in the coordinates of points $D$, $E$, and $F$ after they have undergone a translation. The given translation rule applied to the points seems to result in:

• Point D: Originally at $(-4, -5)$ translated to $(2, 4)$.
• Point E: Originally at $(-7, -4)$ translated to $(-1, 5)$.
• Point F: Originally at $(-8, -1)$ translated to $(-2, 8)$.

These translations match the "Correct Answer" provided in the image.

Part 2 of 3 (Translation Rule)

The task is to write a general rule for this translation in the form $(x, y) \to (x + a, y + b)$, where $a$ and $b$ represent the shift in the $x$- and $y$-coordinates, respectively.

Observing the translations:

• For $D$: $-4 \to 2$ and $-5 \to 4$. This implies an $x$-shift of $+6$ and a $y$-shift of $+9$.
• For $E$: $-7 \to -1$ and $-4 \to 5$, confirming the $x$-shift of $+6$ and $y$-shift of $+9$.
• For $F$: $-8 \to -2$ and $-1 \to 8$, also confirming the $x$-shift of $+6$ and $y$-shift of $+9$.

So, the general translation rule is: $(x, y) \to (x + 6, y + 9)$

This would be the correct selection from the options provided.

Would you like further details on this solution, or do you have additional questions?

Here are some related questions you might find useful:

1. What are translations, and how are they applied in geometry?
2. How can you determine the translation vector from a given set of original and translated points?
3. What is the significance of each component in a translation rule, like $(x + a, y + b)$?
4. How can translations be combined with other transformations, such as rotations or reflections?
5. Can translations change the shape or size of geometric figures?

Tip: When solving transformations, always compare each coordinate shift individually to ensure accuracy in determining the translation vector.