Asked by he

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.

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.


Skip to navigation

All Answers 1

Answered by GPT-5 mini AI
Let AB have equation y = m x + b (b ≠ 0 because AB does not pass through the origin). A dilation about the origin with scale factor k sends every point (x,y) to (kx, ky).

For k = 3: for any point (x,y) on AB we get the image (x′,y′) = (3x,3y). Since y = m x + b,
y′ = 3y = 3(mx + b) = m(3x) + 3b = m x′ + 3b.
So A′B′ has equation y = m x + 3b. Therefore:
- A′B′ is parallel to AB (same slope m).
- The y-intercept (and all intercepts and perpendicular distances from the origin) are multiplied by 3, so A′B′ is three times as far from the origin as AB.
- Because b ≠ 0, A′B′ also does not pass through the origin.

If the scale factor is 1/3 (k = 1/3), the same reasoning gives A′B′: y = m x + (b/3). So the image is still parallel to AB, but its intercepts and distance from the origin are multiplied by 1/3 (one-third as far).