To determine the scale factor used in the dilation of line segment AB, we first need to identify the coordinates of points A and B and their corresponding points A' and B' after dilation.
The points are:
- A (1, -3)
- B (2, -4)
- A' (4, -12)
- B' (8, -16)
The scale factor (k) for a dilation centered at the origin can be calculated using the formula:
\[ k = \frac{\text{distance from origin to A'}}{\text{distance from origin to A}} \]
First, we calculate the distances from the origin to each point. The distance from the origin to a point \( (x, y) \) is given by the formula:
\[ d = \sqrt{x^2 + y^2} \]
Calculating the distance from the origin to point A: \[ d_A = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
Calculating the distance from the origin to point A': \[ d_{A'} = \sqrt{4^2 + (-12)^2} = \sqrt{16 + 144} = \sqrt{160} \]
Now, we compute the scale factor k: \[ k = \frac{d_{A'}}{d_A} = \frac{\sqrt{160}}{\sqrt{10}} = \sqrt{\frac{160}{10}} = \sqrt{16} = 4 \]
Therefore, the scale factor used in the dilation of line segment AB, centered at the origin, is 4.