Results 1 to 2 of 2

Math Help - Transformations

  1. #1
    Member
    Joined
    Mar 2010
    Posts
    144

    Transformations

    Show that a translation TPQ is equivalent to the composition of two reflections in parallel lines l1 and l2, where l1 and l2 are perpendicular to PQ. If d is the distance between l1 and l2, show that PQ = 2d.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Mar 2010
    Posts
    144
    This is what I have so far. I feel like I am missing some inportant parts.

    Proof. Given translation TPQ, pick an arbitrary point, A, on PQ. Construct a line to PQ through A. We shall call this line l1. Reflect point P in l1 to get P'. By the property of reflections, PA = P'A. Let PA = x. Therefore, P'A = x also. Now construct the bisector of P'Q. We shall call this new line, l2. Since l1 is perpendicular to PQ and l2 is perpendicular to PQ, l1 || l2. Let the intersection of l2 and PQ be at point B. Now reflect P' in l2 to get P''. By the property of reflections, P'' will be on Q. Also by the properties of reflections, P'B = P''B = QB. Let the P'B = y. Therefore, P''B = y and QB = y also. Therefore, PQ = x + x + y + y = 2x + 2y and AB (the distance between l1 and l2) = x + y. Let d = AB (the distance between l1 and l2). Therefore, 2d = 2(x + y). Therefore, PQ = x + x + y + y = 2x + 2y = 2(x + y) = 2d. Therefore PQ = 2d.
    Given l1 || l2, and an arbitrary point P. Reflect P over l1 to get P'. Construct line segment PP' where A is the intersection of l1 and PP'. By properties of reflections, PA = P'A. Let PA = x. Therefore, P'A = x also. Next, reflect P' in l2. Construct line segment P'P" where B is the intersection of l2 and P'P". By properties of reflections, P'B = P"B. Let the P'B = y. Therefore, P''B = y also. Therefore, PP" = x + x + y + y = 2x + 2y and AB (the distance between l1 and l2) = x + y. Let d = AB (the distance between l1 and l2). Therefore, 2d = 2(x + y). Therefore, PQ = x + x + y + y = 2x + 2y = 2(x + y) = 2d. Therefore PQ = 2d.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Transformations
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: May 9th 2010, 06:21 PM
  2. Transformations of e^x and lnx
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: March 16th 2010, 05:42 PM
  3. Transformations
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 18th 2009, 10:47 AM
  4. transformations
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 1st 2008, 10:17 AM
  5. f(x) transformations
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: April 11th 2008, 12:46 PM

Search Tags


/mathhelpforum @mathhelpforum