Q&A #3411


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From: Pat Ballew (for Teacher2Teacher Service)
Date: Mar 19, 2000 at 16:23:44
Subject: Re: Locus

  Most of these require thinking through the information and making some
logical deductions. I can walk you through these, but doing these in general
is a matter of experience.  A "do more, get better" kind of situation.  There
just aren't any shortcuts for them. I have laced some comments into the
questions below.

>1.  Given a segment AB, construct and label the locus of points at a distance
AB from point A and equidistant from A and B.

   The set of all points any common distance from another point is a circle,
so "locus of points at a distance AB from point A" means a circle with a
radius of AB; but the fact that they have to be "equidistant from A and B."
means that you also need a circle centered at B with a radius of AB.  The two
points where these intersect are the locus of points that meet the

>1.  In a plane, the locus of the centers of all circles with radius 3cm that
are tangent to a given line L

  A circle can be tangent to a line on either side, so all the circles on one
side would have their centers in a straight line 3cm from the line... those
on the opposite side would have their centers 3 cm from the line on the other
side, together they would form a pair of parallel lines 6 cm apart.

>2.  In a plane, the locus of points 4cm from the center of a circle whose
radius is 5cm.
>   This one is similar to the first, the locus of points  4 cm from the
center is a circle of radius 4 cm.  It will be concentric (same center) as
the first and be a little smaller.  Together the two circles form what is
called an annulus (like annual rings of a tree).

>I am homeschooling my son and these problems are giving him a difficult
time, I donít know much about math, history is my skill.

Most geometry books have several of these in them, and they often seem
strange to students because the language is antiquated and formal.  Try
playing with these with your son.  You make one up and ask him to figure it
out, then let him make one up for you.  You can probably both come up with
some that are more difficult than either he, you, or I can solve, so make a
rule that you can't ask it, if you can't answer it.  Here are some easy ones
to start on:
In two dimensions:
  -The set of all points one inch from a line
  -The set of all points the same distance from two points
  -The set of all points equidistant from two lines
And in three dimensions:
  -All points one inch above the floor
  -The set of all points three inches from a line (this is different than in
two space.)

 I hope some of this helps. Good luck

 -Pat Ballew, for the Teacher2Teacher service

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