When line \( AB \) undergoes dilation from a center of dilation at the origin (0,0), several effects take place based on the scale factor applied.
Effects of Dilation with Scale Factor of 3:
-
Length/Distance: Each point \( A \) and \( B \) on line \( AB \) is moved outward from the origin by a factor of 3. Thus, the distance between any two points on line \( A'B' \) will be three times that of line \( AB \).
-
Position: If a point \( A(x_1, y_1) \) on line \( AB \) is dilated, the new coordinates \( A' \) of the translated point will be \( A'(3x_1, 3y_1) \). The same applies to point \( B \) which will become \( B'(3x_2, 3y_2) \).
-
Orientation/Shape: The slope of line \( AB \) remains unchanged, meaning the angle that line \( AB \) makes with the axes is preserved. This results in line \( A'B' \) remaining parallel to line \( AB \).
-
No intersection with the origin: Since line \( AB \) does not pass through the origin and is simply scaled away from it, line \( A'B' \) will also not pass through the origin.
Effects of Dilation with Scale Factor of 13:
If the scale factor changes from 3 to 13, the overall effects remain similar, but the intensity of the transformation increases:
-
Length/Distance: The distance between points on line \( A'B' \) will now be 13 times greater than that of line \( AB \).
-
Position: The coordinates of each point will be scaled by a factor of 13, meaning if \( A(x_1, y_1) \) becomes \( A'(13x_1, 13y_1) \) and \( B'(13x_2, 13y_2) \).
-
Orientation/Shape: The slope remains unchanged as before. The angle that line \( AB \) makes with the axes is preserved, thus line \( A'B' \) is still parallel to line \( AB \).
-
No intersection with the origin: Similar to the previous case, line \( A'B' \) will also not pass through the origin.
Summary:
- Both dilations expand line \( AB \) outwards from the origin while preserving its slope and not intersecting the origin. However, the dilation with a scale factor of 13 results in a significantly larger scaled line than with a factor of 3.
1 answer