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 13
, 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 k be the scale factor and the center be the origin. A dilation multiplies every point (x,y) on AB by k to get (kx, ky).

For k = 3:
- A' and B' have coordinates (3x_A, 3y_A) and (3x_B, 3y_B).
- The slope is unchanged: slope(A'B') = (3y_B−3y_A)/(3x_B−3x_A) = (y_B−y_A)/(x_B−x_A). So A'B' is parallel to AB.
- If AB has equation y = m x + c (with c ≠ 0 because AB does not pass through the origin), the image has equation y = m x + 3c (the y‑intercept is multiplied by 3).
- The perpendicular distance from the origin to the line is multiplied by 3 (distance |c|/√(m^2+1) becomes |3c|/√(m^2+1)).

For k = 13:
- All the same conclusions hold, with 13 in place of 3: A' and B' are (13x,13y), A'B' is parallel to AB, the y‑intercept becomes 13c, and the distance from the origin is multiplied by 13.