To determine the scale factor of the dilation of line segment AB centered at the origin, we can use the coordinates of the points A, B, A', and B'.
The original points are:
- A (1, -3)
- B (2, -4)
And the dilated points are:
- A' (4, -12)
- B' (8, -16)
The scale factor \( k \) can be found by comparing the coordinates of the original points with the dilated points.
For point A: \[ k = \frac{x'}{x} = \frac{4}{1} = 4 \] \[ k = \frac{y'}{y} = \frac{-12}{-3} = 4 \]
For point B: \[ k = \frac{x'}{x} = \frac{8}{2} = 4 \] \[ k = \frac{y'}{y} = \frac{-16}{-4} = 4 \]
Since the scale factor is consistent for both points, we confirm that the scale factor used in the dilation is:
\[ \text{Scale factor} = 4 \]
Thus, the correct response is 4.
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