To determine the scale factor used in the dilation of line segment \( AB \) with a center of dilation at the origin, we can consider the coordinates of the points before and after the dilation.
Let's define the points:
- Point \( A \) (before dilation): \( A_1 (-3, 4) \)
- Point \( B \) (before dilation): \( B_1 (2, -4) \)
- Point \( A' \) (after dilation): \( A_2 (-6, 8) \)
- Point \( B' \) (after dilation): \( B_2 (4, -8) \)
The scale factor \( k \) can be calculated by comparing the coordinates of the original points and the dilated points. The formula for dilation about the origin is:
\[ (x', y') = (k \cdot x, k \cdot y) \]
We can find \( k \) for each point:
-
For point \( A \):
- Original point \( A_1 (-3, 4) \)
- Dilated point \( A_2 (-6, 8) \)
Applying the dilation formula: \[ -6 = k \cdot (-3) \implies k = \frac{-6}{-3} = 2 \] \[ 8 = k \cdot 4 \implies k = \frac{8}{4} = 2 \]
Thus, \( k = 2 \) for point \( A \).
-
For point \( B \):
- Original point \( B_1 (2, -4) \)
- Dilated point \( B_2 (4, -8) \)
Applying the dilation formula: \[ 4 = k \cdot 2 \implies k = \frac{4}{2} = 2 \] \[ -8 = k \cdot (-4) \implies k = \frac{-8}{-4} = 2 \]
Thus, \( k = 2 \) for point \( B \).
Since the scale factor \( k \) is consistent for both points before and after dilation, we conclude that the scale factor for the dilation of line segment \( AB \) is
\[ \boxed{2}. \]