Let's analyze the concept of dilation in the context of a line segment in a coordinate plane. When we dilate a line segment, we are effectively scaling the distances of the points on the line segment from a specific center of dilation (often the origin).
Dilation with a Scale Factor of \(3\):
When line AB is dilated with a scale factor of \(3\), the coordinates of points A and B, let's say \(A(x_1, y_1)\) and \(B(x_2, y_2)\), are transformed as follows:
\[ A' = (3x_1, 3y_1) \] \[ B' = (3x_2, 3y_2) \]
The resulting line segment A'B' is parallel to AB because dilation preserves the angles and the orientation of the lines. Additionally, because the scale factor is \(3\), the distance from the origin to both points A' and B' is \(3\) times the distances to points A and B, respectively. Thus, the new line A'B' is indeed parallel to AB and \(3\) times farther from the origin.
Dilation with a Scale Factor of \(\frac{1}{3}\):
Now, when we perform a dilation with a scale factor of \(\frac{1}{3}\), the coordinates transform in a similar way:
\[ A' = \left(\frac{x_1}{3}, \frac{y_1}{3}\right) \] \[ B' = \left(\frac{x_2}{3}, \frac{y_2}{3}\right) \]
In this case, the segments A'B' remain parallel to line AB, but since the scale factor is \(\frac{1}{3}\), the distances from the origin to points A' and B' are \(\frac{1}{3}\) of the distances to points A and B. Therefore, A'B' is parallel to AB and \(\frac{1}{3}\) times as far from the origin.
Summary:
- With scale factor \(3\): A'B' is parallel to AB and \(3\) times farther from the origin.
- With scale factor \(\frac{1}{3}\): A'B' is parallel to AB and \(\frac{1}{3}\) times as far from the origin.
This illustrates how dilation affects both the coordinates of points on a line and their relative distances from a point of reference, while preserving parallelism.