Question
Dilation with a Scale Factor of 3:
So, if you have any point \( (x, y) \) on line AB, after dilation, it gets stretched to \( (3x, 3y) \). This means every point moves further away from the origin (0,0) by a factor of 3.
The line still has the same slope, so it keeps its angle, but it shifts to a new position. And because line AB doesn't go through the origin, the new line, which we can call line A'B', won't go through the origin either.
Overall, every point on the line is now 3 times further from the origin than it was before.
Dilation with a Scale Factor of 13:
This is similar to the first case, but now we’re stretching the points even more. Each point \( (x, y) \) on line AB will transform into \( (13x, 13y) \).
This means the line A'B' stretches away from the origin much more than with the scale factor of 3. The slope stays the same, so the line remains parallel to the original line AB, but again, it won’t go through the origin either.
The distance from the origin to each point on this new line is now 13 times as far as it was before!
In Summary:
Both of these dilations keep the same slope for line AB, which means the new lines (A'B' for both scale factors) are going to be parallel to the original line.
The main difference is how much further away the new lines are from the origin: with a scale factor of 3, they’re 3 times as far, and with a scale factor of 13, they’re 13 times as far.
So, dilation is all about stretching away from the origin while keeping the line’s slope, and the scale factor just tells us how much further from the origin the new line ends up!
make this less energetic and show more of the work
Answers
GPT-4o mini
### Dilation with a Scale Factor of 3:
When we perform dilation with a scale factor of 3 on a point \( (x, y) \) located on line AB, we apply the dilation transformation as follows:
\[
(x, y) \rightarrow (3x, 3y)
\]
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 \( (3x, 3y) \), are located farther away from the origin \( (0,0) \) 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 \( (3x, 3y) \), 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 \( (x, y) \) on line AB, the points transform according to the following equation:
\[
(x, y) \rightarrow (13x, 13y)
\]
In this case, each point is stretched even further away from the origin compared to the previous dilation. The new coordinates \( (13x, 13y) \) 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.
### Summary
To 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.
When we perform dilation with a scale factor of 3 on a point \( (x, y) \) located on line AB, we apply the dilation transformation as follows:
\[
(x, y) \rightarrow (3x, 3y)
\]
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 \( (3x, 3y) \), are located farther away from the origin \( (0,0) \) 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 \( (3x, 3y) \), 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 \( (x, y) \) on line AB, the points transform according to the following equation:
\[
(x, y) \rightarrow (13x, 13y)
\]
In this case, each point is stretched even further away from the origin compared to the previous dilation. The new coordinates \( (13x, 13y) \) 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.
### Summary
To 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.