To solve the problem of dilating segment $AB$ using a scale factor of 2 and the origin $(0, 0)$ as the center of dilation, we need to perform the following steps:

1. Identify the coordinates of points $A$ and $B$:

   • From the image, the coordinates of point $A$ are $(-4, 2)$.
   • The coordinates of point $B$ are $(2, -4)$.

2. Apply the dilation transformation with a scale factor of 2:

   • For dilation with the center at the origin $(0, 0)$, the new coordinates of each point after dilation are given by multiplying both the $x$-coordinate and the $y$-coordinate by the scale factor.

   • For point $A$ at $(-4, 2)$: $A' = (2 \times (-4), 2 \times 2) = (-8, 4)$

   • For point $B$ at $(2, -4)$: $B' = (2 \times 2, 2 \times (-4)) = (4, -8)$

3. Record the new coordinates:

   • The new coordinates of point $A'$ are $(-8, 4)$.
   • The new coordinates of point $B'$ are $(4, -8)$.

So, after the dilation, the coordinates of points $A'$ and $B'$ are:

• $A' = (-8, 4)$
• $B' = (4, -8)$

Let me know if you would like a detailed explanation or have further questions!

Here are some questions to expand on this:

1. How would the dilation change if the scale factor was 0.5?
2. What are the properties of dilation in terms of preserving shapes?
3. How can we generalize the dilation process for any point $P(x, y)$?
4. How does changing the center of dilation affect the final coordinates?
5. What other transformations can we perform on this figure, such as rotation or reflection?

Tip: In a dilation, the distances between the points and the center of dilation change proportionally to the scale factor, but angles and relative proportions remain the same!