To determine the scale factor used in the dilation of line segment AB centered at the origin, we can denote the scale factor by \( k \).
If the original points A and B have coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \), respectively, then after dilation by a factor of \( k \) centered at the origin, the new points A' and B' will have coordinates \( (k \cdot x_1, k \cdot y_1) \) and \( (k \cdot x_2, k \cdot y_2) \).
To find the scale factor, we typically need to compare the lengths of the original line segment AB and the dilated line segment A'B'. The length of a line segment from point A to point B is calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
The length after dilation (A'B') will then be:
\[ d' = \sqrt{(k \cdot x_2 - k \cdot x_1)^2 + (k \cdot y_2 - k \cdot y_1)^2} = k \cdot \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = k \cdot d \]
The scale factor \( k \) describes how much longer or shorter the line segment becomes due to dilation relative to its original length.
Without specific coordinates for points A and B or lengths, we can't determine \( k \) directly. However, among the provided response options—\( 2, -4, 4, \frac{1}{4} \)—the scale factor must be a positive value to maintain the orientation, unless specified otherwise.
If we need to assume the most common acceptable values of typical dilations:
- \( 2 \) indicates the segment doubled in length,
- \( 4 \) means the segment quadrupled,
- \( \frac{1}{4} \) means the segment is one-fourth its original length.
The negative options indicate a reversal in orientation which may not be typical unless explicitly stated.
To summarize without coordinates or lengths, the choice can be guided by common dilation behavior. A responsible assumption in typical scenarios could likely be one of the positive values, such as \( 2 \), \( 4 \), or \( \frac{1}{4} \).
If you have particular numerical lengths or positions in mind to compute, please provide that, and we can find the specific \( k \).