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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/The Pythagorean Theorem">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{The Pythagorean Theorem\label{Pythagorean theorem}}

The Pythagorean theorem is given as the 47th proposition in book 1 of
Euclid's \textit{Elements}. The Pythagoreans (6th century B.C.) probably
gave the first general proof of the theorem. However, long before the time
of Pythagoras, statements closely related to the Pythagorean theorem and its
converse were made in Egypt, Babylonia, China, and India.

\begin{theorem}
(\emph{Pythagorean Theorem}) The area of the square upon the hypotenuse of a
right triangle is equal to the sum of the areas of the squares on the other
two sides.\bigskip \FRAME{dtbpF}{1.5359in}{1.6821in}{0pt}{}{}{pythag1.wmf}{%
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\end{theorem}

This theorem is often stated as the equation 
\[
c^{2}=a^{2}+b^{2} 
\]
where $c$ is the length of the\label{Hypotenuse} 
\CustomNote[hypotenuse]{Margin Hint}{%
The hypotenuse of a right triangle is the side opposite the right angle.} of
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The converse of the Pythagorean theorem is also true:

\begin{theorem}
If $c^{2}=a^{2}+b^{2}$, then the triangle is a right triangle with
hypotenuse of length $c$.\bigskip
\end{theorem}

\begin{example}
$5^{2}=3^{2}+4^{2}$ and a triangle with sides of lengths $5$, $3$, and $4$
units is a right triangle.\bigskip
\end{example}

\ \ The Pythagorean theorem is applied widely, because it leads directly to
a formula for the distance between two points.

\subsection{\ Related Reference Topics}

{\small \hyperref{Algebra}{}{}{Al_contents.tex}\hfill }

{\small \hyperref{Trigonometry}{}{}{Tr_contents.tex}}

{\small \hyperref{Table of Fundamental Algebraic Laws and Theorems}{}{}{%
Tab_algebra_rules.tex}}

{\small \hyperref{Table of Formulas For Length, Area, and Volume}{}{}{%
TAB_geom_formulas.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Angular Measure}{}{}{../../Help/DMIndex.tex#Angular measure%
}}

{\small \hyperref{Conic Sections}{}{}{../../Help/DMIndex.tex#Conic sections}}

{\small \hyperref{Plotting Curves and Surfaces}{}{}{../../Help/DM6.tex}}

{\small \hyperref{Polygons}{}{}{../../Help/DMIndex.tex#Polygons}}

{\small \hyperref{Surfaces of Revolution}{}{}{%
../../Help/DMIndex.tex#Surfaces of revolution}}

{\small \hyperref{Trigonometry}{}{}{../../Help/DM4.tex}}

\noindent \hrulefill

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