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January 19th, 2016, 03:01 PM  #1 
Newbie Joined: Mar 2014 Posts: 7 Thanks: 0  shortest distance from a point to a line?
This is about the shortest distance from the point P to the line (see figure 1). I'm trying to prove it is given by the length of the segment perpendicular to the line that joins the line to the point. Is this argument I give 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. Is this correct? Last edited by dante; January 19th, 2016 at 03:07 PM. 
January 20th, 2016, 10:37 PM  #2 
Senior Member Joined: Nov 2010 From: Indonesia Posts: 1,176 Thanks: 111 
Sounds good to me. Dunno about the others, though.

January 21st, 2016, 06:46 AM  #3 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 9,425 Thanks: 640 
Another way: Let P be center of circle radius PQ. Draw line RQ tangent to the circle. Conclusion? 
January 22nd, 2016, 06:51 AM  #4  
Newbie Joined: Mar 2014 Posts: 7 Thanks: 0  Quote:
Right so, the circle is touching the line at one point (namely the point Q) and the line is tangent to the circle. You use that as the line is tangent to a circle, the circle radius PQ will be perpendicular to the line. The segment PR starts at the center and extends outside the circle and so is longer than the radius of the circle, and hence longer than the (perpendicular) segment PQ. Last edited by dante; January 22nd, 2016 at 07:11 AM.  
January 22nd, 2016, 07:47 AM  #5 
Newbie Joined: Mar 2014 Posts: 7 Thanks: 0 
Here is the diagram.


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