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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
\ /
Q
/ \
/   \
/     \

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
\   .
Q
.   \
.       \
\

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
http://mathforum.org/dr.math/

```

```
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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