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?

Answer 1

When you dilate about the origin by a positive scale factor k, every point (x,y) on the original line goes to (kx,ky). For a line AB that does not pass through the origin this does the following:

- The image A′B′ is parallel to AB (slope is unchanged).
- Every point on AB moves radially away from the origin by factor k, so A′ and B′ lie on the same rays from the origin as A and B.
- Intercepts (x- and y-intercepts) and the perpendicular distance from the origin are multiplied by k.
- Algebraically: if AB is y = m x + b (b ≠ 0), then A′B′ is y = m x + k b. If AB is ax + by + c = 0, then A′B′ is a x + b y + k c = 0, so the distance from the origin becomes k times the original distance.

For k = 3: A′B′ is parallel to AB, has the same slope, and all distances from the origin (and intercepts) are 3 times those of AB.

If the scale factor were 13 instead, everything is the same except every distance/intercept is multiplied by 13 instead of 3 (A′B′ still parallel to AB, same slope).