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.

3
The answer is mostly correct. The student made small errors while solving.

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

Most work is shown or explained.

2
The answer is partially correct. The student made errors while solving.

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

Some work is shown or explained.

1
The answer is incorrect. The student made errors while solving.

The answer shows little problem-solving skills or reasoning.

Little or no work is shown.

0 The question is not answered.


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Answer 1

A dilation centered at the origin with scale factor 3 sends each point (x,y) on line AB to (3x,3y). If AB has equation y = m x + b (b ≠ 0 because AB does not pass through the origin), the image A′B′ is the set of points (X,Y) = (3x,3y) with Y/3 = m(X/3) + b. Simplifying gives the equation of A′B′:

Y = mX + 3b.

Conclusions (scale factor 3)
- Slope: unchanged (m). So A′B′ is parallel to AB.
- y-intercept: multiplied by 3 (from b to 3b), so A′B′ does not pass through the origin (since b ≠ 0).
- Distance from the origin to the line: multiplied by 3 (distance formula for y = mx + b gives |b|/√(m^2+1), which becomes |3b|/√(m^2+1)).
- Every point and every segment on AB is moved radially away from the origin by a factor of 3.

If the scale factor were 13, the same reasoning applies: the image line would be y = m x + 13b, so it would still be parallel to AB, have the same slope, have its intercept and perpendicular distance from the origin multiplied by 13, and not pass through the origin.