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Distance between Point and Line

Date: 02/08/2009 at 10:47:27
From: Lucy
Subject: Distance from lines

What distance is the point (1,2) from the line 3x + 6y = 10?

I know how to draw the line and how to plot points but I don't know 
how to find the closest distance in between points.

Date: 02/11/2009 at 09:35:21
From: Doctor Ian
Subject: Re: Distance from lines

Hi Lucy,

Let's start there, then.  Suppose you plot the line A,

  3x + 6y = 10

and the point P, at (1,2).  Now draw any line B through P that will
intersect A at point Q.  You might get something like

   \             /
    \           /
     \ A       P
      \       /
       \     /
        \   / B
         \ /
         / \
        /   \
       /     \

So far, so good? 

Now, for any line you choose, there will be some distance from P to Q.
There is ONE line that will give you the SHORTEST possible distance.
Which one?  The one that is perpendicular to A.  

     \ A                  P
      \               .
       \           .
        \       .   B
         \   .
       .   \
    .       \

Make sure you understand WHY this is true before going on.  If it's
not clear, let me know. 

In the case where we have the perpendicular line, the distance from P
to Q is considered to BE the distance from the point to the line--
because there is no other possible distance that could be smaller. 

If that all makes sense, then you can rephrase the problem this way:

   If A is the line 3x + 6y = 10,
   and P is the point (1,2),
   and B is a line perpendicular to A and passing through P, 
   find the distance from P to the point where A intersects B. 

Here's something you'll want to know:  If two lines are perpendicular
(and neither is vertical), the product of their slopes is -1.

Is this enough to get started?  Let me know if you need more help.

- Doctor Ian, The Math Forum

Date: 02/11/2009 at 21:12:39
From: Lucy
Subject: Thank you (Distance from lines)

Thank you so much for your help.  Your explanation was perfect and my
understanding is complete.
Associated Topics:
High School Coordinate Plane Geometry
High School Linear Equations

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