To determine the scale factor used in the dilation of line segment AB centered at the origin, we first need to identify the coordinates of the points A and B.
Given:
- Point A = (2, 3)
- Point B = (4, 1)
The length of the original segment AB can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Calculating the distance of the segment AB:
\[ d_{AB} = \sqrt{(4 - 2)^2 + (1 - 3)^2} \] \[ = \sqrt{(2)^2 + (-2)^2} \] \[ = \sqrt{4 + 4} \] \[ = \sqrt{8} \] \[ = 2\sqrt{2} \]
Next, we need a point that represents the dilated version of AB. Let's say for instance the new points after dilation are A' = (6, 9) and B' = (12, 3).
We will compute the distances for the new line segment A'B':
\[ d_{A'B'} = \sqrt{(12 - 6)^2 + (3 - 9)^2} \] \[ = \sqrt{(6)^2 + (-6)^2} \] \[ = \sqrt{36 + 36} \] \[ = \sqrt{72} \] \[ = 6\sqrt{2} \]
Now we can find the scale factor \( k \) using the formula:
\[ k = \frac{d_{A'B'}}{d_{AB}} = \frac{6\sqrt{2}}{2\sqrt{2}} = \frac{6}{2} = 3 \]
Thus, the scale factor used in the dilation of line segment AB is 3.