To determine the scale factor used in the dilation of line segment AB centered at the origin, we need to find the original distance between points A and B and the distance after dilation.
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Find the distance between points A and B:
The coordinates of points A and B are:
- A(1, -3)
- B(2, -4)
We can use the distance formula to find the distance \(d\) between points A and B: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Plugging in the coordinates: \[ d = \sqrt{(2 - 1)^2 + (-4 + 3)^2} \] \[ = \sqrt{(1)^2 + (-1)^2} \] \[ = \sqrt{1 + 1} \] \[ = \sqrt{2} \]
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Assuming a dilation from the origin:
After dilation by a scale factor \(k\), the new coordinates of A and B (denoted as \(A'\) and \(B'\)) will be:
- \(A'(k \cdot 1, k \cdot -3)\) → \(A'(k, -3k)\)
- \(B'(k \cdot 2, k \cdot -4)\) → \(B'(2k, -4k)\)
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Calculate the distance after dilation:
Using the new coordinates: \[ d' = \sqrt{(2k - k)^2 + (-4k + 3k)^2} \] \[ = \sqrt{(k)^2 + (-k)^2} \] \[ = \sqrt{k^2 + k^2} \] \[ = \sqrt{2k^2} \] \[ = k\sqrt{2} \]
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Determine the scale factor:
The scale factor \(k\) can be found by comparing the new distance \(d'\) to the original distance \(d\): \[ \frac{d'}{d} = \frac{k\sqrt{2}}{\sqrt{2}} = k \]
Given the options provided (2, 4, 1/4, -4), we don't have a specified new distance or a specific value for \(k\).
But knowing that dilation factors are generally positive and if \(k\) is a valid dilation factor, its value must be positive.
As no additional information is provided to determine a specific dilation or if it denotes a specific context where it must conform to one of the given options, a conclusive scale factor is not derivable from the provided information.
If we had a specific new length after dilation or additional parameters, we could solve for \(k\).
Based on typical choice constraints, you would typically select 2 or 4 as plausible options, assuming some standard dilation occurs. However, without specific criteria, I suggest validating against additional context. In dilation, the factor is usually positive and increasing distances should be observed.
Conclusion: More information is required to determine a specific scale factor from the provided options.