Question

What happeneds when the center of dilation is on a line and then you dilate the line?

Answer 1

Answer: Nothing happens

The line does not change.

Consider the line y = 2x. It goes through (0,0) and (1,2).

Dilate both points by some scale factor. Let's say we go with k = 5.

(0,0) is the center of dilation and won't move. But (1,2) moves to (5,10).

But notice how (5,10) is also on the line y = 2x. All we've done is shift one point around on this same line. The line itself does not move at all. If we were dealing with a line segment then the segment would shrink or grow depending if the scale factor is 0 < k < 1 or k > 1 respectively.

But in the case of a line itself (not a segment), it goes on infinitely in both directions. The idea of growing or shrinking something infinitely long doesn't make sense. This is why there isn't any change.

Desmos can be used to verify this.

Answer 2

The line is shortened

Explanation:

If the center of dilation is on the line, then it is shrinking along the line, meaning it will be shortened.