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?(4 points)



Math Short Answer Rubric (4 points)
This table shows the criteria for awarding points and the corresponding point values.
Points Awarded Criteria
4
The answer is clear and correct.

The answer shows that the student used problem-solving skills and reasoning.

All work is shown or explained.

Answer 1

Let line AB have equation y = m x + b with b ≠ 0 (since AB does not pass through the origin). A dilation about the origin with scale factor k sends each point (x,y) on AB to (x',y') = (k x, k y). Then

y' = k y = k(m x + b) = m(k x) + k b = m x' + k b.

So the image line has equation y' = m x' + k b.

Effects (k = 3):
- The slope m is unchanged, so A′B′ is parallel to AB.
- The y-intercept is multiplied by 3 (becomes 3b), so the line moves away from the origin; the perpendicular distance from the origin to the line is multiplied by 3.
- Every point of AB is carried along the ray from the origin through that point to a point three times as far from the origin.

If k = 1/3:
- Same conclusions with k = 1/3: slope unchanged (A′B′ still parallel to AB).
- The y-intercept becomes b/3, so the line moves closer to the origin; perpendicular distances to the origin are multiplied by 1/3.
- Every point moves along its ray to a point one-third as far from the origin.

(If the original line passed through the origin, it would map to itself; that does not occur here because b ≠ 0.)