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Drawing a Locus of Points

Date: 05/30/2008 at 00:00:22
From: Erin
Subject: Geometry- Drawing a Locus

I was doing my geometry homework, and some of the questions are 
rather difficult and some are easy. For example: 

All points in the plane that are equidistant from the rays of an 
angle and equidistant from two points on one side of the angle.

The directions say to drew a diagram to find the locus of points that 
satisfy the conditions. 

I got the first part before the word "and." My drawing:

   /
  /  
 / 
/_______ 

Okay, so there's the angle, and then I drew in an angle bisector which
I was unable to do on here. 

Another problem I had trouble with was:
 
All points in space equidistant from two intersecting planes. 

I drew two planes intersecting perpendicularly just to make it 
easier to visualize. 
                            ______
                            |    |
                       ___o |    |o____
                       |    |    |    |
                       |____|    |____|
                          o |____|o
The dots are the four points equidistant from each of the planes, but 
the book shows that where the dots are, are two other planes that 
intersect each other. And the four planes intersect at one point.

Overall, do you have any tips in reading a complicated problem and 
drawing it?

I think my major problem with these locus problems, is reading the 
problem. I just don't think I understand what I'm reading, or I may 
understand part of it and when the second part comes, I don't know 
what to do. I think I'm having problems channeling all the 
information it gives, so I try to break it up. When I break it up, I 
sometimes have problems putting it together.

My teacher told me to split the problem into two, like do the first 
part before "and" then do the second part, which would be after "and."
He also told us to draw the given in one color, then the points or 
whatever it is that the problem says in another color, so where the 
non-given things that meet is what we describe.

I know that the locus is the place where things meet.



Date: 05/30/2008 at 23:23:16
From: Doctor Peterson
Subject: Re: Geometry- Drawing a Locus

Hi, Erin.

You've asked a big question; but let's see if I can help one part at a
time, then come to some broader conclusions.

You wrote:

>I was doing my geometry homework, and some of the questions are 
>rather difficult and some are easy. For example: 
>
>All points in the plane that are equidistant from the rays of an 
>angle and equidistant from two points on one side of the angle.
>
>The directions say to drew a diagram to find the locus of points that 
>satisfy the conditions. 
>
>I got the first part before the word "and."
>My drawing:
>   /
>  /  
> / 
>/_______ Okay, so there's the angle, and then I drew in an angle 
>bisector which I was unable to do on here. 

You are given two conditions:

  A: equidistant from the rays (sides) of the angle
  B: equidistant from two (given) points on one side (ray)

To find all points that satisfy BOTH A and B, you can draw each locus
(A and B) separately, and the intersection of those will be your
combined locus--the only point(s) for which both A and B are true.

You've drawn locus A, a ray from the vertex.  Now let's add in two
given points on the horizontal side:

          /
         /
        /
       /
      /        / <--- locus A
     /      /
    /   /
   / /
  o-----p-------q---

What is the locus B, the points that are the same distance from p as
from q?  You're probably as familiar with this one as with the angle
bisector.  Draw it, and where that line crosses locus A is the single
point in the locus "A and B".


>Another problem I had trouble with was:
> 
>All points in space equidistant from two intersecting planes. 
>
>I drew two planes intersecting perpendicularly just to make it 
>easier to visualize. 
>                            ______
>                            |    |
>                       ___o |    |o____
>                       |    |    |    |
>                       |____|    |____|
>                          o |____|o
>The dots are the four points equidistant from each of the planes, but 
>the book shows that where the dots are, are two other planes that 
>intersect each other. And the four planes intersect at one point.

It can be quite difficult to draw something like this in the first
place; I wouldn't say that your planes look like they intersect, but
perhaps you can visualize what you have in mind anyway.  Here's how I
might draw it (the best I can in plain text, and I'm pretty much an
expert at it!):

               +
              /|
             / |
            +  |
            |  |
     +------|  +--------+
    /       | /:       /
   /        |/ :      /
  +---------+--------+
            |  |
            |  +
            | /
            |/
            +

I drew the planes so they look perpendicular, which they will not
generally be; but it's hard enough to draw them this way!

Now, where can you be and be the same distance from both planes?  The
situation is much like the angle bisector of an angle in a plane; in
fact, this figure represents a "dihedral angle" (between two planes).
A side view (still making them look perpendicular) would be:

            ^
    .       |       .
      .     |     .
        .   |   .
          . | .
  <---------+-------->
          . | .
        .   |   .
      .     |     .
    .       |       .
            v

The two dotted lines represent all points in the plane that are
equidistant from the two solid lines.  Now imagine this as just a side
view of the planes; the dotted lines, too, are side views of planes,
which pass through the same line of intersection.  I'll just barely
try to draw this:

               +
       .      /|       .
      .  .   / |     ..
     .     .+  |   . .
    .       |  | .  .
     +.-----|  +--.-----+
    /   .   | /:.      /
   /      . |/.:      /
  +---------+--------+
       .  . | .|       .
      . .   |  +.     .
     ..     | /   .  .
    .       |/      .
            +

Note that, since it's hard both to draw and to visualize three
dimensions, I used two dimensions in two ways: first, using an ANALOG
(intersecting lines in a plane are LIKE intersecting planes in space,
so they give a sense of what the result might be like); and second,
using a SIDE VIEW (or cross section) in order to draw part of what I'm
imagining, still keeping things simpler than trying to picture the
whole thing.  We do the same thing to try to visualize 
four-dimensional objects, which is quite a trick!


>I think my major problem with these locus problems, is reading the 
>problem. I just don't think I understand what I'm reading, or I may 
>understand part of it and when the second part comes, I don't know 
>what to do. I think I'm having problems channeling all the 
>information it gives, so I try to break it up. When I break it up, I 
>sometimes have problems putting it together.

That's not a bad analysis.  Reading mathematicians' twisted grammar
just takes time and experience!  A big part of math is being able to
break a large problem into pieces, and then be able to put them back
together.  Often that skill is developed by being even more orderly
than you need to be once you understand the topic more fully; you may
need to write down more of what you are doing (sort of like making
notes to yourself to remind yourself where you left your glasses when
you took them off, so you can find them again later).  For example, I
gave names to the two parts, A and B, so I could clearly communicate
with myself (and you) about those parts.  I first summarized the
overall strategy (draw A, draw B, intersect), and then filled in the
outline.  Such self-communication (naming things and writing down what
they mean) and organization (outlining a plan before doing it) is very
helpful.  You won't always see ahead of time what all the steps will
be, but you can at least keep good records of what you have done while
you explore.  Think of yourself as Lewis and Clark, not knowing just
how they'd get to the Pacific, but making their own maps so they'd
know where they'd been once they found where they were!  Problem-
solving in general requires that sort of skill.  But I'm digressing...


>My teacher told me to split the problem into two, like do the first 
>part before "and" then do the second part, which would be after
>"and." He also told us to draw the given in one color, then the
>points or whatever it is that the problem says in another color, so
>where the non-given things that meet is what we describe.

All of that is good advice.  (I used dotted lines for a similar
purpose.)  It just takes time and practice to make it all work!

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 05/31/2008 at 13:01:50
From: Erin
Subject: Thank you (Geometry- Drawing a Locus)

Thank you very much for answering my question. I understand it better,
and I think I'll do more problems over the weekend and try to get
better at it. Thanks for the Lewis and Clark advice and organization
advice. I'll start being more organized again. I think organization
helps a lot in math! Thanks again!
Associated Topics:
High School Euclidean/Plane Geometry
High School Higher-Dimensional Geometry

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