To determine the scale factor used in the dilation of the line segment \( AB \) (from points \( A(1, -3) \) to \( B(2, -4) \)) to its corresponding points \( A'(4, -12) \) and \( B'(8, -16) \), we can analyze the coordinates.
Step 1: Calculate the lengths of the segments
Length of Segment \( AB \): Using the distance formula between points \( A(x_1, y_1) \) and \( B(x_2, y_2) \): \[ d_{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - 1)^2 + (-4 + 3)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
Length of Segment \( A'B' \): Now, calculate the distance for points \( A'(4, -12) \) and \( B'(8, -16) \): \[ d_{A'B'} = \sqrt{(8 - 4)^2 + (-16 + 12)^2} = \sqrt{(4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
Step 2: Determine the scale factor
The scale factor \( k \) is the ratio of the lengths of the segments: \[ k = \frac{d_{A'B'}}{d_{AB}} = \frac{4\sqrt{2}}{\sqrt{2}} = 4 \]
Thus, the scale factor used in the dilation of line segment \( AB \), centered at the origin, is 4.
Final Answer:
4