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Describe the Locus


Date: 03/28/2002 at 02:01:00
From: Andrea
Subject: Geometry

Describe in words the locus that satisfies the given condition.  What 
is the locus of all points in a plane two inches from a point A?

I took the basic information from the book regarding this, but I 
don't know if this is correct because they haven't given me any 
condition.

RESOLUTION:  The locus is the circle with the center P and radius of 
1cm.  The locus is the sphere with the center P and radius of 2".

Thanks for your help; it is greatly appreciated.


Date: 03/28/2002 at 12:33:15
From: Doctor Peterson
Subject: Re: Geometry

Hi, Andrea.

I'm not sure what you mean by saying they haven't given a condition; 
the condition is that the points are 2 inches from A.

Since you are in a plane, the answer can't be a sphere. But a circle 
with radius 2 inches (not 1 centimeter!) will fit the description.

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


Date: 03/28/2002 at 21:45:07
From: Andrea
Subject: Geometry

Sorry, I can't draw very well with the computer yet. It is actually 
a circle with an x at the top of the circle and the number two is 
inside on the left-hand side.  The degrees are located outside the 
circle.  Hope I'm explaining this right.  Thanks.


Date: 03/28/2002 at 22:53:23
From: Doctor Peterson
Subject: Re: Geometry

Hi, Andrea.

Actually, it's better to describe a geometrical figure in words than 
to draw it - thinking about how to say it often helps you understand 
the problem better.

Here's my version:

                    ooooooooooooo
               o+ooo            +ooooo
             oo   \           /       oo
          ooo       \       /           ooo
      40 o            \   /                o
       oo           2   X                   oo 60
      o              /    \                   o
      o            /        \                 o
     o           /            \                o
    o          /                \               o
    o        /                    \             o
    o      /                        \           o
    o    /                            \         o
    o  /                                \       o
     +                                    \    o
      o                                     \ o
      o                                       +
       oo                                   oo
         o                                 o
          ooo                           ooo
             oo                       oo
               ooooo             ooooo
                    ooooooooooooo

I presume the 40 and 60 are the measures of the arcs, that is, their 
central angles. The formula here is

    angle 2 = (40 + 60)/2

If you forget this, you can draw an extra line and use the simpler 
fact that an inscribed angle is half the central angle of the arc it 
subtends:

                    ooooooooooooo
               o+ooo            +ooooo
             oo   \           /       oo
          ooo       \       /           ooo
         o            \   /                o
     a oo           x   X                   oo
      o              /    \                   o b
      o            /        \                 o
     o           /            \                o
    o          /                \               o
    o        /                    \             o
    o      /                        \           o
    o    /                            \         o
    o  /  z                             \       o
    A+--------________                    \    o
      o               -------________   y   \ o
      o                              ---------+B
       oo                                   oo
         o                                 o
          ooo                           ooo
             oo                       oo
               ooooo             ooooo
                    ooooooooooooo

Then y = a/2 and z = b/2, and x, an exterior angle of triangle AXB, is 
the sum, x = y+z = (a+b)/2.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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