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Symmetric of a point relative to a line

  1. Dec 24, 2011 #1
    1. The problem statement, all variables and given/known data
    What is the easiest way of finding the symmetrical of a point relative to an arbitrary line?
    (I was asked on an exam to find the symmetrical of a point relative to the line y = x, but that's rather trivial - just switch the coordinates. How can I do it for any arbitrary line ax + by = c?)

    2. Relevant equations

    3. The attempt at a solution
    I found a way, but it's rather messy. Given the line s and the point P, find the line t that is perpendicular to s and passes through P. Calculate the distance d between P and s. Find the point on the line t that is also at a distance d from s.

    This sounds awfully complicated and messy for me. Is there a quicker way?
  2. jcsd
  3. Dec 25, 2011 #2


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    Staff: Mentor

    That's the way I'd do it. I don't know of any other.
  4. Dec 25, 2011 #3
    For a line of the form Ax+By+C = 0, distance d from a point p is:

    d = |Ax+By+C|/(A2+B2)1/2

    [|...| is simply the absolute value meant to keep d>/=0]
  5. Dec 25, 2011 #4
    This is precisely the formula I would use in the description I have given. But it's rather
    cumbersome to use it twice in a single problem. Is there a way around?
  6. Dec 26, 2011 #5
    Why would you use it twice? Using it once will provide you with the shortest distance from the line to the point.
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