To answer this question, we need to understand what happens to a line segment under a dilation transformation. Dilation is a transformation that changes the size of an object without altering its shape. It is defined by a scale factor and a center of dilation.
Given a line segment and a center of dilation that is not on the line segment, when we dilate this line segment by a scale factor of 2, the following will occur:
1. The length of the line segment will be scaled by the factor of 2. This means that the length of the dilated image of the line segment will be twice the length of the original line segment.
2. The direction of the line segment will remain the same because dilation does not affect angles or parallelism. This means that the dilated image of the line segment will be parallel to the original line segment.
Putting these two pieces of information together, we can rule out options (1) and (3), since the line segments will not be perpendicular after a dilation. We can also rule out option (4) because the length of the image will not be one-half of the original; it will be twice as long due to the scale factor of 2.
Therefore, the correct answer is option (2): The line segments are parallel, and the image is twice the length of the given line segment.
