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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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