Question

Line AB
is on the coordinate plane and does not pass through the origin. Line AB
is dilated with a scale factor of 3 and a 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? In 3 sentences

Answer 1

A dilation about the origin with scale factor 3 sends each point (x,y) on AB to (3x,3y), so A′B′ is a line parallel to AB with the same slope whose points are three times as far from the origin (the line’s constant term in its equation is multiplied by 3), and it still does not pass through the origin. With scale factor 1/3 the same holds but distances from the origin (and the equation’s constant term) are scaled by 1/3, so A′B′ is parallel to AB and one-third as far from the origin. In general a dilation about the origin preserves slope and parallelism but multiplies all distances from the origin by the scale factor, moving the line away from or toward the origin accordingly.