Question
line AB is on the coordinate plane and does not pass through the origin. line AB is dilated to scale factor of 3 and center dilation at the center of dilation at the origin to create line A' B'. describe the effects of the dilation on line AB. If line AB was dilated with a scale factor of 1/3 how would your answer change
Answers
Answered by
GPT-4o mini
To understand the effects of dilation on line AB and how it affects the coordinates, let's first define the two scenarios you've described.
### Dilation with a Scale Factor of 3
1. **Dilation Description**: When line AB is dilated with a scale factor of 3 from the origin, every point \( (x, y) \) on line AB is transformed to a new point \( (3x, 3y) \). This means that the line shifts away from the origin; all points on line AB are moved three times farther from the origin.
2. **Effects on Line Properties**:
- **Length**: The dilation will increase the length of the line segment AB by a factor of 3. If the distance between points A and B is \( d \), the distance between points A' and B' after dilation will be \( 3d \).
- **Slope**: The slope of line AB will remain the same after dilation because the ratio of the vertical and horizontal distances (which defines the slope) does not change.
- **Position**: The entire line segment is expanded outward from the origin, but its orientation and slope stay constant.
### Dilation with a Scale Factor of 1/3
1. **Dilation Description**: Conversely, if line AB is dilated with a scale factor of \( \frac{1}{3} \) from the origin, every point \( (x, y) \) on line AB is transformed to a new point \( \left(\frac{x}{3}, \frac{y}{3}\right) \). This transformation brings each point closer to the origin.
2. **Effects on Line Properties**:
- **Length**: The dilation will decrease the length of the line segment AB by a factor of \( \frac{1}{3} \). Thus, if the original distance between points A and B was \( d \), the distance between points A' and B' will be \( \frac{1}{3}d \).
- **Slope**: Just like with the scale factor of 3, the slope will remain unchanged. The orientation of the line remains the same, but the line segment gets shorter.
- **Position**: The segment shrinks towards the origin, with all points moving closer to it.
### Summary
- **Scale Factor of 3**: Line AB expands outward from the origin, increasing in length by a factor of 3, while maintaining the same slope and orientation.
- **Scale Factor of 1/3**: Line AB contracts towards the origin, decreasing in length by a factor of \( \frac{1}{3} \), while also retaining its slope and orientation.
In conclusion, both dilations maintain the slope of the line but affect the length and position in opposite ways—one elongates it, while the other shortens it.
### Dilation with a Scale Factor of 3
1. **Dilation Description**: When line AB is dilated with a scale factor of 3 from the origin, every point \( (x, y) \) on line AB is transformed to a new point \( (3x, 3y) \). This means that the line shifts away from the origin; all points on line AB are moved three times farther from the origin.
2. **Effects on Line Properties**:
- **Length**: The dilation will increase the length of the line segment AB by a factor of 3. If the distance between points A and B is \( d \), the distance between points A' and B' after dilation will be \( 3d \).
- **Slope**: The slope of line AB will remain the same after dilation because the ratio of the vertical and horizontal distances (which defines the slope) does not change.
- **Position**: The entire line segment is expanded outward from the origin, but its orientation and slope stay constant.
### Dilation with a Scale Factor of 1/3
1. **Dilation Description**: Conversely, if line AB is dilated with a scale factor of \( \frac{1}{3} \) from the origin, every point \( (x, y) \) on line AB is transformed to a new point \( \left(\frac{x}{3}, \frac{y}{3}\right) \). This transformation brings each point closer to the origin.
2. **Effects on Line Properties**:
- **Length**: The dilation will decrease the length of the line segment AB by a factor of \( \frac{1}{3} \). Thus, if the original distance between points A and B was \( d \), the distance between points A' and B' will be \( \frac{1}{3}d \).
- **Slope**: Just like with the scale factor of 3, the slope will remain unchanged. The orientation of the line remains the same, but the line segment gets shorter.
- **Position**: The segment shrinks towards the origin, with all points moving closer to it.
### Summary
- **Scale Factor of 3**: Line AB expands outward from the origin, increasing in length by a factor of 3, while maintaining the same slope and orientation.
- **Scale Factor of 1/3**: Line AB contracts towards the origin, decreasing in length by a factor of \( \frac{1}{3} \), while also retaining its slope and orientation.
In conclusion, both dilations maintain the slope of the line but affect the length and position in opposite ways—one elongates it, while the other shortens it.