Thread: Isometry Preserves Straight Lines Proof?

1. Isometry Preserves Straight Lines Proof?

I'm asked to prove as a corollary to: the image of the line segment PQ under F is a line segment between F(P) and F(Q), that an isometry preserves straight lines.

Here is the proof I've written down:
Let F be an isometry.
Let F(P) be denoted by P'.
Let P,Q be arbitrary points on a line L. Let X be a point on the line segment PQ.

Since F preserves distances we know that: d(P,X)=d(P',X') and d(Q,X)=d(Q',X').
We have d(P,Q)=d(P,X)+d(X,Q).
By assumption of F preserving distance we also have d(P',Q')=d(P',X')+d(X',Q').

Thus the image of segment PQ is contained in its image, segment P'Q'; but since P,Q are arbitrary, the segment PQ may be considered indefinitely long. Thus the line L is mapped to a straight line L', since the arbitrarily large segments composing that line are straight.

Is it correct? Am I writing too much? Is there a better way? Am I missing a second part? ie. to prove to sets equal you must show each belongs to the other.

2. Originally Posted by Ultros88
I'm asked to prove as a corollary to: the image of the line segment PQ under F is a line segment between F(P) and F(Q), that an isometry preserves straight lines. Is it correct? Am I writing too much? Is there a better way?
It is difficult to give a definite answer to your questions because definitions and theorem orders differ. But generally you work is correct. You have shown that a line segment is mapped into an line segment. You may want to expand to include the entire line.

Recall that if three points, P Q & X, are collinear then P-X-Q, P-Q-X, or X-P-Q.
You have done the first, the segment. The others follow from what you have done.

3. How to Expand Proof to be for a line?

How can I expand it to include a line if all I know is that a line is determined by two points? It seems to me that I can only deal with segments, or rays I guess. But again, it seems to me to boil down to proving that arbitrary segments on the oppositely directed rays, say XP and XQ, are straight. hmm...?

Plato, from your reply it seems like there is some other way to prove the lines are straight. I'm guessing by taking 3 arbitrary collinear points and proving as I have already done. Is that right? How do the other two cases determine the entire line?

Thanks,
Ultros

4. As I said the first time, I do not know your set of axioms nor do I know the sequence of theorems. However, in general it seems to me as if you have proved betweeness for P-X-Q. All I meant is, I would want you to show that isometry also preserves the relations X-P-Q and P-Q-X. Then you have considered the entire line. But again your course may require a different level of rigor, more or less.