When line \( AB \) is dilated with a scale factor of 3 and a center of dilation at the origin, every point \( (x, y) \) on line \( AB \) will be transformed to a new point \( (3x, 3y) \) on the new line \( A'B' \).
Effects of the Dilation with Scale Factor 3:
- Distance from the Origin: Every point on line \( AB \) is moved away from the origin, becoming three times as far from the origin.
- Line Direction and Slope: The direction (or slope) of the line remains the same since dilation is a uniform scaling that preserves the angle with the x-axis.
- Line Equation: If the equation of line \( AB \) is \( y = mx + b \), the new line \( A'B' \) after dilation will have the equation \( y = mx + b' \), where \( b' \) will change depending on the original \( b \) because the y-intercept \( b \) will be scaled as well.
If the Scale Factor is \( \frac{1}{3} \):
If line \( AB \) were dilated with a scale factor of \( \frac{1}{3} \), the transformation would work differently:
- Distance from the Origin: Each point would be moved closer to the origin, becoming one-third as far from it.
- Line Direction and Slope: As before, the line's slope would remain unchanged, preserving the angle.
- Line Equation: The equation of line \( A'B' \) would still keep the slope \( m \), but the y-intercept \( b \) would also be scaled down. For instance, if the original line had the equation \( y = mx + b \), the new line would have the equation \( y = mx + b' \) where \( b' \) is one-third of the original \( b \).
In summary, the dilation caused by a scale factor of 3 expands the line away from the origin, while a scale factor of \( \frac{1}{3} \) contracts the line towards the origin, but in both cases, the slope remains constant.