When line AB is dilated with a scale factor of 4 and a center of dilation at the origin, every point \( P(x, y) \) on line AB is transformed to point \( P'(4x, 4y) \). This means that the coordinates of each point on line AB are scaled by a factor of 4 away from the origin.
Effects of the dilation with a scale factor of 4:
- Relative Location: The dilated line A'B' will remain parallel to the original line AB, as the direction and slope of the line do not change during dilation.
- Size: Line A'B' will be four times larger than line AB. This scaling means that the distances between points on line A'B' will be four times those of corresponding points on line AB.
Because the dilation is centered at the origin and the line does not pass through the origin, the entire line A'B' will appear stretched out, moving away from the origin while keeping the same orientation.
Effects of a dilation with a scale factor of 1/4:
If line AB were dilated by a scale factor of 1/4 from the origin, each point \( P(x, y) \) on line AB would be transformed to \( P'(\frac{1}{4}x, \frac{1}{4}y) \).
- Relative Location: Similar to the previous scenario, line A'B' would still be parallel to line AB, as the orientation is unaffected by dilation.
- Size: Line A'B' would be one-fourth the length of line AB. Thus, the distances between points on line A'B' will be one-fourth those of corresponding points on line AB.
Summary:
- Dilation by scale factor 4: Line A'B' is larger (4 times the length of line AB) and remains parallel, positioned further from the origin.
- Dilation by scale factor 1/4: Line A'B' is smaller (1/4 the length of line AB), remains parallel, and is closer to the origin.
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