Show lines segments parallel? Based on the attached, if $AB$ and $DC$ are parallel, then $$\frac{|OD|}{|OA|}=\frac{|OC|}{|OB|}=\frac{|CD|}{|AB|}$$ Using the theorem, how can I show the following when the image is dilated by a dilation factor of $r$:

The line segments go to parallel line segments

I am not sure what this question is asking. I thought it was saying to show that $rAB$ and $rDC$ are parallel (so kind of the theorem in reverse). If so, can I show that the lines are parallel proceeding by contradiction and assuming they're not? So then, if the lines aren't parallel, then they must intersect at some point, $p$? Any insight is appreciated.

Thanks

If we dilate everything by a factor of $r$, then the ratios $$\frac{|OD|}{|OA|}=\frac{|OC|}{|OB|}=\frac{|CD|}{|AB|}$$ will continue to hold, right? Have you learned that that is a sufficient condition for $AB$ and $CD$ to be parallel?
In particular, you can show that $OAB$ and $ODC$ are similar triangles (try Side angle side), and therefore angle $OAB$ and angle $ODC$ are the same.