To determine the scale factor used in the dilation of line segment \(AB\) (from points \(A(1, -3)\) and \(B(2, -4)\)) to points \(A'(4, -12)\) and \(B'(8, -16)\), we can use the formula for dilation centered at the origin:
\[ k = \frac{(x', y')}{(x, y)} \]
where \(k\) is the scale factor, \((x, y)\) are the original coordinates, and \((x', y')\) are the coordinates after dilation.
Let's calculate the scale factor for point \(A\):
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For point \(A(1, -3)\) and \(A'(4, -12)\):
- The coordinates of \(A\) are \((1, -3)\) and of \(A'\) are \((4, -12)\).
For the x-coordinates: \[ k_x = \frac{4}{1} = 4 \]
For the y-coordinates: \[ k_y = \frac{-12}{-3} = 4 \]
Since both calculations give \(k = 4\), the scale factor for point \(A\) is \(4\).
Next, let's check point \(B\):
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For point \(B(2, -4)\) and \(B'(8, -16)\):
- The coordinates of \(B\) are \((2, -4)\) and the coordinates of \(B'\) are \((8, -16)\).
For the x-coordinates: \[ k_x = \frac{8}{2} = 4 \]
For the y-coordinates: \[ k_y = \frac{-16}{-4} = 4 \]
Again, both calculations give \(k = 4\).
Thus, the scale factor used in the dilation of line segment \(AB\) centered at the origin is:
\[ \boxed{4} \]