Question

Line AB
A
B
is on the coordinate plane and does not pass through the origin. Line AB
A
B
is dilated with a scale factor of 3 and a center of dilation at the origin to create line A′B′
A
′
B
′
. Describe the effects of the dilation on Line AB
A
B
. If line AB
A
B
was dilated with a scale factor of 13
1
3
, how would your answer change?(

Answer 1

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:
1. **Distance from the Origin**: Every point on line \( AB \) is moved away from the origin, becoming three times as far from the origin.
2. **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.
3. **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:
1. **Distance from the Origin**: Each point would be moved closer to the origin, becoming one-third as far from it.
2. **Line Direction and Slope**: As before, the line's slope would remain unchanged, preserving the angle.
3. **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.