\hfill \hfill } %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.4in,right=0.5in,top=0.5in,bottom=0.4in} \pagestyle{fancy} \lhead{\color{blue} \Large Lecture Notes} \chead{\color{black} \LARGE Area - Part 3} \rhead{ page \ \thepage} \cfoot{} \lfoot{\small \copyright $\;$ Hidegkuti, 2009} \rfoot{\small Last revised: December 20, 2013} \textwidth 7.5in \textheight 9.6in \setlength{\headheight}{27pt} \setlength{\parindent}{0in} \begin{document} \begin{center} {\LARGE Part 3 - General Triangles\bigskip } \end{center} Let us now consider general triangles.\medskip %TCIMACRO{% %\TeXButton{box start - 7.5in}{\begin{boxedminipage}{7.5in} %\setlength{\fboxrule}{15pt} %\setlength{\fboxsep}{15pt}}}% %BeginExpansion \begin{boxedminipage}{7.5in} \setlength{\fboxrule}{15pt} \setlength{\fboxsep}{15pt}% %EndExpansion %TCIMACRO{\TeXButton{white}{\color{white}}}% %BeginExpansion \color{white}% %EndExpansion .% %TCIMACRO{\TeXButton{black}{\color{black}}}% %BeginExpansion \color{black}% %EndExpansion \textbf{Theorem:} \ The area of a general triangle with sides $a,$ $b,$ $c$ and height $h$ as shown on the picture below is $A=\dfrac{ah}{2}$.\FRAME{% dtbpF}{1.9043in}{1.2782in}{0pt}{}{}{insert4b.bmp}{\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 1.9043in;height 1.2782in;depth 0pt;original-width 1.4728in;original-height 1.0533in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert4b.bmp';file-properties "XNPEU";}} %TCIMACRO{\TeXButton{end of box}{\end{boxedminipage}}}% %BeginExpansion \end{boxedminipage}% %EndExpansion Proof: \ As before, we will use a previously obtained result. Since the general triangle no longer has a right angle, we create it by drawing in the altitude or height belonging to the side $a.$ Now we split our triangle into two right triangles, and each of them is half of a rectangle.\FRAME{dtbpF}{% 2.4068in}{1.6898in}{0pt}{}{}{insert5.bmp}{\special{language "Scientific Word";type "GRAPHIC";display "USEDEF";valid_file "F";width 2.4068in;height 1.6898in;depth 0pt;original-width 1.8334in;original-height 1.3604in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert5.bmp';file-properties "XNPEU";}}Our triangle makes up for half of a rectangle, with sides $a$ and $h$. Thus $A=\dfrac{ah}{2}$.\bigskip \textit{Example 3:} \ Find the area of the triangle shown on the picture below.\FRAME{dtbpF}{1.7876in}{1.3327in}{0pt}{}{}{insert5b.bmp}{\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.7876in;height 1.3327in;depth 0pt;original-width 1.8334in;original-height 1.3604in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'insert5b.bmp';file-properties "XNPEU";}}Solution: It is important to notice that we will not need all the information given. We apply the area-formula.% \begin{equation*} A=\dfrac{ah}{2}=\dfrac{21\unit{in}\left( 12\unit{in}\right) }{2}=\dfrac{252% \unit{in}^{2}}{2}=126\unit{in}^{2} \end{equation*} \bigskip \bigskip \bigskip \begin{center} {\LARGE Practice Problems\bigskip } \end{center} \begin{enumerate} \item Find the perimeter and area of the triangle shown on the picture below. \ Include units in your computation and answer. \FRAME{dtbpF}{2.3212in% }{1.3162in}{0pt}{}{}{insert.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.3212in;height 1.3162in;depth 0pt;original-width 2.2814in;original-height 1.2816in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../../../../ALGEBRA/prealgebra_problemsets/problemsets21_30/ps26/insert.bmp';file-properties "XNPEU";}% } \item Find the perimeter and area of the triangle shown on the picture below. $\ $Include units in your computation and answer.\FRAME{dtbpF}{% 2.3212in}{1.3162in}{0pt}{}{}{insert.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 2.3212in;height 1.3162in;depth 0pt;original-width 2.2814in;original-height 1.2816in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '../../../../../ALGEBRA/prealgebra_problemsets/problemsets21_30/ps27/insert.bmp';file-properties "XNPEU";}% } \end{enumerate} \bigskip \bigskip \begin{center} {\LARGE Practice Problems - Answers\bigskip } \end{center} \bigskip \begin{enumerate} \item $P=34\unit{in}~~~~~~~A=52\unit{in}^{2}$ \item $P=48\unit{ft}~~~~~A=\allowbreak 84\unit{ft}^{2}$\vspace{3in} \end{enumerate} \href{http://www.teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html% }{For more documents like this, visit our page at\ http://www.teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}