To analyze the problem of dilation where a line segment is dilated by a scale factor, let's carefully examine what happens:
### Definition and Properties of Dilation
- Dilation: It's a transformation that scales an object by a certain factor with respect to a center of dilation.
- Scale Factor: The ratio by which the object is scaled. In this case, it is given as 2, meaning the image will be twice the size of the original object.
- Center of Dilation: The fixed point around which the dilation occurs. It's given that this point is not on the line segment.
### Key Points:
1. When dilating a line segment by a scale factor around a center not on the line, the slopes of the original segment and its dilated image are unchanged.
2. Since the slopes remain the same, the two line segments (original and dilated) will be parallel.
3. The length of the image will be scaled by the given scale factor. Here, the scale factor is 2, so the length of the dilated line segment will be twice the length of the original.
### Analysis of Statements:
1. The line segments are perpendicular, and the image is one-half of the length of the given line segment.
This statement cannot be true because dilation preserves the direction of the line segments (hence, they remain parallel), and it states that the image is one-half of the length when the scale factor is actually 2, not 0.5.
2. The line segments are perpendicular, and the image is twice the length of the given line segment.
This statement cannot be true because, as mentioned above, dilation does not change the direction of line segments (they remain parallel).
3. The line segments are parallel, and the image is twice the length of the given line segment.
This statement is true. The line segments remain parallel, and the image will be twice the length of the given line segment due to the scale factor of 2.
4. The line segments are parallel, and the image is one-half of the length of the given line segment.
This statement cannot be true because, although the line segments will remain parallel, the given scale factor is 2 which makes the image twice as long, not one-half.
### Conclusion:
The correct statement is:
3) The line segments are parallel, and the image is twice the length of the given line segment.
