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A check: shortest distance from point to line

  1. Jan 19, 2016 #1

    julian

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    We want to know the shortest distance from the point P to the line (see figure 1). As far as I know it is given by the length of the segment perpendicular to the line that joins the line to the point. Can you check this argument I give is correct?

    Part A. First let us draw in the segment from the point P to the line that meets the line at 90 degrees (makes a right angle). We call this the perpendicular segment.

    We call the point where the perpendicular segment meets the line Q (see figure 2).

    Part B. IMPORTANT!:

    We prove that the perpendicular segment represents the shortest distance from the point to the line by demonstrating that ANY OTHER SEGMENT from the point P to the line is longer!

    Part C. To that end consider any point other than Q on the line, call it R. (see figure 3)

    Part D. We draw in the segment from the point P to the point R.

    We notice that the points P,Q, and R are the corners of a right angled triangle where the segment from P to R is the hypotenuse and the perpendicular segment (from P to Q) is one of the other sides (see figure 4).

    Part E. It is well known that the hypotenuse of a right angled triangle is the longest side. Thus we have proved that ANY OTHER SEGMENT is longer than the perpendicular segment.

    Proof complete.
     

    Attached Files:

  2. jcsd
  3. Jan 19, 2016 #2

    PhanthomJay

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    nicely and simply stated.
    you mean Q. E. D. (look it up if you are under 50 years young)!
     
  4. Jan 21, 2016 #3

    julian

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