To solve this, we first need to determine the translation rule from the given information. We know that the point $(2, -9)$ translates to $(4, -14)$.

Step 1: Determine the translation vector.

The translation vector is the change in the coordinates from the original point to the new point:

• Change in x-coordinate: $4 - 2 = 2$
• Change in y-coordinate: $-14 - (-9) = -14 + 9 = -5$

So, the translation vector is $(2, -5)$.

Step 2: Apply the translation to the point $(5, 6)$.

Now, we'll apply the same translation vector $(2, -5)$ to the point $(5, 6)$:

• New x-coordinate: $5 + 2 = 7$
• New y-coordinate: $6 + (-5) = 6 - 5 = 1$

Final Answer:

The image of the point $(5, 6)$ under the same translation is $(7, 1)$.

Would you like any further clarification or details on this?

Related questions:

1. What is a translation in geometry?
2. How do you find the translation vector between two points?
3. What happens to a figure when it undergoes translation?
4. Can a translation change the shape or size of a figure?
5. How do different types of transformations (translation, rotation, reflection) differ?

Tip: When working with translations, always find the vector first by subtracting the coordinates of the original point from the translated point.