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 multiply each coordinate of points \( A \) and \( B \) by the scale factor.
The coordinates of the endpoints are:
- \( A(3, 7) \)
- \( B(4, 9) \)
Now apply the dilation:
For point \( A \): \[ A' = (3 \times 2, 7 \times 2) = (6, 14) \]
For point \( B \): \[ B' = (4 \times 2, 9 \times 2) = (8, 18) \]
Therefore, after the dilation, the coordinates are: \[ A'(6, 14) \text{ and } B'(8, 18) \]
Looking at the provided options, the correct answer is: A′(6,14) and B′(8,18).
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