To determine the scale factor used in the dilation of line segment AB centered at the origin, we first need to find the lengths of the original segment and the dilated segment.
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Calculate the length of segment AB for the first pair of points A(1, -3) and B(2, -4):
Length of AB = √[(x₂ - x₁)² + (y₂ - y₁)²]
Where (x₁, y₁) = A(1, -3) and (x₂, y₂) = B(2, -4).
Length of AB = √[(2 - 1)² + (-4 + 3)²] = √[(1)² + (-1)²] = √[1 + 1] = √2.
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Calculate the length of segment AB for the second pair of points A(4, -12) and B(8, -16):
Length of AB = √[(8 - 4)² + (-16 + 12)²]
Length of AB = √[(4)² + (-4)²] = √[16 + 16] = √32 = 4√2.
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Now, we find the scale factor (k):
Scale factor (k) = Length of dilated segment / Length of original segment = (4√2) / (√2) = 4.
Thus, the scale factor used in the dilation of line segment AB is 4.