Presentation

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04/Solution #04/Presentation.pdf Executable file

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%%Tema para beamer "Imunam", versión 1.0
\documentclass{beamer}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[spanish]{babel}
\graphicspath{ {images/} }
\usepackage{mathtools}
\usepackage{tikz}
%% Logos de restaurantes:
\newcommand{\logomanekin}{images/logomanekin}
%%Se define el "environment" teorema
%% The "environment" theorem is defined
\newtheorem{Formula}{Formula}
%%Tema de beamer "Imunam"
%% Theme of beamer "Imunam"
\usetheme[cuernavaca]{Imunam}
%\usetheme{Imunam}
%%Si se omite "[cuernavaca]" en éste comando, el logotipo se imprime sin la
%%leyenda "Unidad Cuernavaca" en la parte inferior.
%% If "[cuernavaca]" is omitted in this command, the logo is printed without the
%% legend "Cuernavaca Unit" in the lower part.
\title{Proof of the Pythagoras theorem.}
\author{Enrique Andrade Gonzalez %Nombre del autor % Author's name
\texttt{e.andrade@udc.es}} %email
\date{Computer tools in mathematican's work} %subject
%\institute{Instituto de Matemáticas, unidad Cuernavaca}
%%Instituto del ponenete, dado que el texto "Intituto de matemáticas" aparece
%%en el logo, parece redundante incluirlo además con éste comando.
\begin{document}
\begin{frame}
\titlepage %Necesario para generar la portada
\end{frame}
%%La siguiente diapositiva es opcional, si se quiere la tabla de contenidos
%%Se sebe compilar dos veces el documento para que funcione
%--------------------------------------------------------------------------
\begin{frame}
\tableofcontents %Imprime la tabla de contenido
\end{frame}
%--------------------------------------------------------------------------
\section{Introduction} %%Título de la sección (Opcional)
\begin{frame}
\frametitle{Introduction}
\framesubtitle{objective} %%Subtítulo de la diapositiva (opcional)
In this presentation we try to show a proof of the Pythagorean theorem.
There are many demonstrations, but this one is one of the simplest.
\end{frame}
\section{Proof of the Pythagorean theorem} %%Otra sección
\begin{frame}
\frametitle{Concept}
Suppose we have a square of side \textbf{r} and on each of its sides we place a right triangle of legs \textbf{x} and \textbf{y}. As in this situation the hypotenuse of each of the triangles is \textbf{r} we want to prove that:
\\\
\begin{Formula} %%Uso del "environment" definido al inicio del documento.
\[x^{2}+y^{2}=r^{2}\]
\end{Formula}
\end{frame}
\begin{frame}
\frametitle{The figure}
The figure that is obtained is the following:
\begin{center}
\begin{tikzpicture}[scale=0.8]
%\begin{tikzpicture}
%\draw (0,0) circle (1cm);
\draw [fill=green] (-2,-2) rectangle (2,2);
\draw [fill=red] (-2,0) -- (0,2) -- (2,0) -- (0,-2) -- cycle;
% r
\draw [dotted]
(0.75,1) node[black] {r}
(0.75,-1) node[black] {r}
(-0.75,1) node[black] {r}
(-0.75,-1) node[black] {r};
% x y
\draw [dotted]
(1,-2.2) node[black] {y}
(1,2.2) node[black] {x}
(-1,2.2) node[black] {y}
(-1,-2.2) node[black] {x}
(2.2,1) node[black] {y}
(2.2,-1) node[black] {x}
(-2.2,1) node[black] {x}
(-2.2,-1) node[black] {y};
\end{tikzpicture}
\[x^{2}+y^{2}=r^{2}\]
\end{center}
\end{frame}
%\section{Restaurants} %%Otra sección
\begin{frame}
\frametitle{Conclusions}
\begin{itemize}
\item Each side of the \textbf{green square} is the sum of \textbf{x} and \textbf{y}. Therefore, the area of the square is: \[(x+y)^{2}\]
\item For the same reason, the area of the \textbf{red square} is: \[r^{2}\]
\item The area of each of the \textbf{green triangles} (y, x and r) is:\[\frac{x+y}{2}\]
\end{itemize}
\end{frame}
\section{Demonstration} %%Otra sección
\begin{frame}
\frametitle{Demonstration}
\begin{itemize}
\item The green square is formed by the red square and the four green triangles, so the sum of all the areas is: \[(x+y)^{2}=r^{2} + 4 (\frac{x+y}{2})\]
\item We develop the left part of equality: \[(x+y)^{2}=x^{2} + 2xy + y^{2}\]
\item We substitute in the first formula: \[x^{2} + 2xy + y^{2} = r^{2} + 2xy\]
\item \textbf{2xy} is eliminated on both sides of the equality, and we obtain the desired result: \[x^{2} + y^{2} = r^{2}\]
\end{itemize}
\end{frame}
\end{document}

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