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Connecting Algebra and Geometry

```Date: 07/11/2002 at 11:58:16
From: Laura Rightmeier
Subject: connection between algebra and geometry

What are the mathematical connections between algebra and geometry? I
have come up with graphing equations and the Pythagorean Theorem, but
that's about it. What else is there?
```

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Date: 07/12/2002 at 08:52:24
From: Doctor Peterson
Subject: Re: connection between algebra and geometry

Hi, Laura.

Just to list some general categories rather than specifics, here are
some connections I see:

1. Geometric formulas: Area, perimeter, etc. are expressed in
algebraic form. The Pythagorean theorem can be put in this category.

2. Algebraic proofs of geometric theorems: Many proofs today use
algebraic manipulation, such as starting with a similarity,
expressing that as an equation, and deriving further equations to
show ultimately that two lengths are equal. Many proofs of the
Pythagorean theorem work this way, and many others use that as one of
formulas are often found by algebra, including calculus (which is

3. Analytic geometry: We can describe a geometric figure in terms of
coordinates, and use that to do proofs in a way very different from
the  previous category. For example, rather than starting with a
similarity, we might start with the equations of two lines and find
their intersection, to do the same proof.

4. Graphing: Going the other way, we can start with an algebraic
problem and represent it graphically, solving an equation by finding
a geometric intersection.

5. Geometric representation of an expression: The Greeks used
geometry to do all their "algebra"; a quadratic equation, for
example, would be discussed as a question about squares and
rectangles rather than powers and products.

If you look through history, as this last item suggests, you will
find that the line between geometry and algebra is often very thin. I
find, in looking at questions we get at Dr. Math, that I often can't
tell whether a student is taking algebra or geometry: is this
question using algebra to find the answer to a geometry problem, or
is it using geometry as the basis for an algebra problem?

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

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