To determine the scale factor used in the dilation of the line segment \( AB \), centered at the origin, we first need to find the length of the original line segment \( AB \) and the length of the dilated line segment.
Step 1: Find the original length of segment \( AB \).
The coordinates of point \( A \) are (2, 3) and the coordinates of point \( B \) are (4, 1). The length of segment \( AB \) is calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ d = \sqrt{(4 - 2)^2 + (1 - 3)^2} \] \[ = \sqrt{(2)^2 + (-2)^2} \] \[ = \sqrt{4 + 4} \] \[ = \sqrt{8} \] \[ = 2\sqrt{2} \]
Step 2: Identify the dilation.
Let's denote the coordinate point after dilation as \( A' \) and \( B' \). When performing a dilation centered at the origin, the coordinates of the points are scaled by a certain factor \( k \). Thus, the coordinates of the dilated points are:
\[ A' = (2k, 3k) \] \[ B' = (4k, 1k) \]
Step 3: Find the length of the dilated segment \( A'B' \).
Applying the distance formula to find \( A'B' \):
\[ d' = \sqrt{(4k - 2k)^2 + (1k - 3k)^2} \] \[ = \sqrt{(2k)^2 + (-2k)^2} \] \[ = \sqrt{4k^2 + 4k^2} \] \[ = \sqrt{8k^2} \] \[ = 2\sqrt{2} |k| \]
Step 4: Determine the scale factor.
Now, we know that the original length \( d \) is \( 2\sqrt{2} \) and the length of the dilated segment \( d' \) is \( 2\sqrt{2} |k| \).
If we set the lengths to be equal (since the question does not specify how the segment is dilated), we have:
\[ 2\sqrt{2} |k| = 2\sqrt{2} \]
To solve for \( |k| \):
\[ |k| = 1 \]
Conclusion
The scale factor \( k \) used in the dilation is:
\[ k = 1 \]
This means that there was no actual dilation and the segment \( AB \) remains unchanged.
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