Question
Dilation with a Scale Factor of 3:When we perform dilation with a scale factor of 3 on a point [lrnwysimath]\( (x, y) \)[/lrnwysimath] located on line AB, we apply the dilation transformation as follows:[lrnwysimath]\[(x, y) \rightarrow (3x, 3y)\][/lrnwysimath]This transformation scales the coordinates of every point on the line by a factor of 3. As a result, the new points, which we can denote as [lrnwysimath]\( (3x, 3y) \)[/lrnwysimath], are located farther away from the origin [lrnwysimath]\( (0,0) \)[/lrnwysimath] than the original points on line AB.Although the distance from the origin has increased, it is important to note that the slope of the line does not change. The line continues to maintain its angle in relation to the x-axis, thereby remaining parallel to the original line AB.It's also worth mentioning that since line AB does not pass through the origin, the new line formed by points [lrnwysimath]\( (3x, 3y) \)[/lrnwysimath], which we can label as line A'B', also does not go through the origin. Thus, the transformation affects the position of the line without changing its orientation.Dilation with a Scale Factor of 13:Following a similar process, if we apply a dilation with a scale factor of 13 to a point [lrnwysimath]\( (x, y) \)[/lrnwysimath] on line AB, the points transform according to the following equation:[lrnwysimath]\[(x, y) \rightarrow (13x, 13y)\][/lrnwysimath]In this case, each point is stretched even further away from the origin compared to the previous dilation. The new coordinates [lrnwysimath]\( (13x, 13y) \)[/lrnwysimath] indicate that these points are now 13 times farther from the origin than their original positions on line AB.Again, while the distance from the origin has changed significantly, the slope of the line remains constant. Therefore, line A'B' will be parallel to line AB, just as it was in the previous dilation.SummaryTo summarize, when performing dilations on line AB:Both dilations preserve the slope of the original line, which indicates that the new lines (A'B' for each scale factor) are parallel to the original line.The difference between the two scenarios lies in the distance from the origin to each point on the new lines:With a scale factor of 3, points are 3 times farther from the origin.With a scale factor of 13, points are 13 times farther.In conclusion, dilation involves increasing the distance of points from the origin while keeping the line’s slope consistent. The scale factor determines how much further away the new line will be positioned.
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Answers
GPT-4o mini
**Dilation with a Scale Factor of 3:**
When dilating a point \( (x, y) \) on line AB by a scale factor of 3, the transformation is:
\[ (x, y) \rightarrow (3x, 3y) \]
This moves the points farther from the origin \( (0,0) \) while maintaining the same slope, meaning the new line A'B' remains parallel to line AB. Since line AB does not pass through the origin, line A'B' also does not go through the origin.
**Dilation with a Scale Factor of 13:**
Similarly, for a scale factor of 13, the transformation is:
\[ (x, y) \rightarrow (13x, 13y) \]
Here, points are stretched even further from the origin, but again, the slope remains unchanged. Line A'B' is still parallel to line AB.
**Summary:**
Both dilations preserve the slope and parallelism of the original line:
- For scale factor 3, points are 3 times farther from the origin.
- For scale factor 13, points are 13 times farther.
Dilation increases the distance from the origin while keeping the slope consistent, with the scale factor determining how much farther new points are located.
When dilating a point \( (x, y) \) on line AB by a scale factor of 3, the transformation is:
\[ (x, y) \rightarrow (3x, 3y) \]
This moves the points farther from the origin \( (0,0) \) while maintaining the same slope, meaning the new line A'B' remains parallel to line AB. Since line AB does not pass through the origin, line A'B' also does not go through the origin.
**Dilation with a Scale Factor of 13:**
Similarly, for a scale factor of 13, the transformation is:
\[ (x, y) \rightarrow (13x, 13y) \]
Here, points are stretched even further from the origin, but again, the slope remains unchanged. Line A'B' is still parallel to line AB.
**Summary:**
Both dilations preserve the slope and parallelism of the original line:
- For scale factor 3, points are 3 times farther from the origin.
- For scale factor 13, points are 13 times farther.
Dilation increases the distance from the origin while keeping the slope consistent, with the scale factor determining how much farther new points are located.