To find the coordinates \( A' \) and \( B' \) after dilation of the line segment \( AB \) with a scale factor of 2, centered at the origin, we can use the following formula for dilation:
\[ (x', y') = (kx, ky) \]
where \( (x, y) \) are the original coordinates, \( (x', y') \) are the new coordinates after dilation, and \( k \) is the scale factor.
Given:
- \( A(3, 7) \)
- \( B(4, 9) \)
- Scale factor \( k = 2 \)
For point \( A \): \[ A' = (2 \cdot 3, 2 \cdot 7) = (6, 14) \]
For point \( B \): \[ B' = (2 \cdot 4, 2 \cdot 9) = (8, 18) \]
Thus, the coordinates after dilation are:
- \( A'(6, 14) \)
- \( B'(8, 18) \)
The correct answer is: A′(6,14) and B′(8,18).