Question
To describe the effects of the dilation on line AB, we need to consider how dilation works in the coordinate plane.
Dilation with Scale Factor of 3:
Each point \( (x, y) \) on line AB is transformed to a new point \( (3x, 3y) \).
This effectively stretches the line away from the origin, maintaining its slope but changing its position. Since line AB does not pass through the origin, line A'B' will also not pass through the origin.
The overall distance from the origin to any point on the line increases by a factor of 3.
Dilation with Scale Factor of 13:
Similar to the previous case, each point \( (x, y) \) on line AB is transformed to \( (13x, 13y) \).
This dilation results in a more significant stretch away from the origin compared to the scale factor of 3. The line A'B' will still maintain its slope and will not pass through the origin.
The overall distance from the origin to any point on the line increases by a factor of 13.
Summary of Changes:
Both dilations maintain the slope of line AB, ensuring that the new lines (A'B' for both scale factors) are parallel to the original line AB.
The primary change is the degree of distance from the origin—the line stretches more significantly with the increased scale factor (13 vs. 3).
So, while the basic characteristics of dilation remain the same (slope preserved and translation away from the origin), the scale factor determines how far the new line is moved away from the origin.
make this sound like a high schooler wrote it
Answers
GPT-4o mini
When we talk about the effects of dilation on line AB in the coordinate plane, it's pretty cool how it works!
1. **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.
2. **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!
1. **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.
2. **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!