Results 1 to 4 of 4

Math Help - Isometry Preserves Straight Lines Proof?

  1. #1
    Junior Member
    Joined
    Mar 2008
    Posts
    40

    Cool 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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,965
    Thanks
    1785
    Awards
    1
    Quote Originally Posted by Ultros88 View Post
    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.

    But again, you must follow your instructor’s standards for rigor.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2008
    Posts
    40

    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
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,965
    Thanks
    1785
    Awards
    1
    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. vector help, straight lines
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: May 7th 2010, 05:02 PM
  2. angle between straight lines
    Posted in the Algebra Forum
    Replies: 2
    Last Post: March 17th 2010, 10:01 AM
  3. Combinatorics - n straight lines?
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 27th 2009, 05:58 AM
  4. Vector - straight lines
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 12th 2008, 11:44 AM
  5. Straight Lines
    Posted in the Math Topics Forum
    Replies: 2
    Last Post: August 30th 2007, 08:06 AM

Search Tags


/mathhelpforum @mathhelpforum