\documentclass[% version=last,% %b5paper,% a5paper, %letterpaper,% 10pt,% %12pt,% headings=small,% bibliography=totoc,% index=totoc,% twoside,% reqno,% cleardoublepage=empty,% open=any,% parskip=half,% %draft=false,% draft=true,% %DIV=classic,% DIV=12,% headinclude=false,% pagesize]%{scrreprt} {scrartcl} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[polutonikogreek,turkish]{babel} \newcommand{\gr}[1]{\selectlanguage{polutonikogreek}#1\selectlanguage{turkish}} \usepackage{pstricks} \usepackage{hfoldsty} \usepackage[neverdecrease]{paralist} \usepackage{amsmath,amsthm} \theoremstyle{definition} \newtheorem{soru}{Soru} \theoremstyle{remark} \newtheorem*{cozum}{\c C\"oz\"um} %\usepackage{verbatim} %\let\cozum=\comment %\let\endcozum=\endcomment \pagestyle{myheadings} \markboth{Mat 113, 1.\ sınav \c c\"oz\"umleri, 2012}{Mat 113, 1.\ sınav \c c\"oz\"umleri, 2012} \begin{document} \title{1.\ sınav \c c\"oz\"umleri} \date{5 Kasım 2012, Saat 15:00} \author{Ahmet Bakkalo\u glu, Ayhan G\"unayd\i n\\ Özer Öztürk, David Pierce} \publishers{MSGS\"U, MAT 113} \maketitle \textbf{A\c c\i klamalar:} %Sınavda 4 sayfada 5 soru var. Lütfen dikkat ederek yazın. \"Oklid'in ilk 20 \"onermesini kullanabilirsiniz. } \textbf{Soru 5.} A\c sa\u g\i daki sorular\i n \emph{biriyi} \c c\"oz\"un. \begin{enumerate}[A.] \item Aşağıdaki bitkilerin ve deniz hayvanlar\i n\i n isimleri Türkçede nedir? \begin{center} \makebox[0cm][c]{\gr{AGKINARA, ASTAKOS, KASTANIA, MUDIA, PRASO, STREIDIA}} \end{center} \item Yunan alfabesini s\i ras\i nda yaz\i n. \item \"Oklid'in ilk 6 \"onermesi i\c cin, kulland\i\u g\i\ \"onermeleri (ve sadece onlar\i) yaz\i n: \end{enumerate} \begin{cozum}\mbox{} \begin{compactenum}[A.] \item Enginar, \i stakoz, kestane, midye, p\i rasa, istiridye. \item \gr{A B G D E Z H J I K L M N X O P R S T U F Q Y W} \item\mbox{} \begin{center} \begin{tabular}{|c|c|} \hline 1&---\\\hline 2&1\\\hline 3&2\\\hline 4&---\\\hline 5&3, 4\\\hline 6&3, 5\\\hline \end{tabular} \end{center} \end{compactenum} \end{cozum} \textbf{Soru 1.} Fig\"urde \begin{compactitem} \item $CBD$, merkezi $A$ olan bir \c cember; \item $CAD$, merkezi $B$ olan bir \c cember; \item $AEB$ ile $CED$, do\u gru \c cizgiler \end{compactitem} olsun. $AEC$ a\c c\i s\i n\i n dik a\c c\i\ oldu\u gunu g\"osterin. \begin{center} \psset{unit=2cm} \begin{pspicture}(3,2) \pscircle(1,1){1} \pscircle(2,1){1} \pspolygon(1,1)(1.5,1.87)(2,1)(1.5,0.13) \psline(1,1)(2,1) \psline(1.5,1.87)(1.5,0.13) \uput[l](1,1){$A$} \uput[r](2,1){$B$} \uput[u](1.5,1.82){$C$} \uput[d](1.5,0.18){$D$} \uput[ur](1.5,1){$E$} \end{pspicture} \end{center} \begin{cozum}\mbox{} \begin{compactenum} \item $AC=AB$ [\c cemberin yar\i\c caplar\i] \item $CB=AB$ [ayn\i] \item $AC=CB$ [1.\ genel kavram] \item $AD=DB$ [benzer] \item $CD$ kenar\i, $ACD$ ve $BCD$ \"u\c cgenlerinin ortak kenar\i d\i r. \item $ACD=BCD$ [8.\ \"onerme] \item $CE$ kenar\i, $ACE$ ve $BCE$ \"u\c cgenlerinin ortak kenar\i d\i r. \item $AEC=BEC$ [4.\ \"onerme] \end{compactenum} \end{cozum} \textbf{Soru 2.} Fig\"urde \begin{compactitem} \item $AB=DE$, $BC=EF$, $AC=DF$, $BG=EH$, $AG=KH$; \item $BGA$ ile $EHK$ a\c c\i lar\i, dik \end{compactitem} olsun. Fig\"ur\"un imk\^ans\i z oldu\u gunu g\"osterin, yani $K$ ile $D$ noktalar\i n\i n farkl\i\ noktalar olamad\i\u g\i n\i\ g\"osterin. \begin{center} \begin{pspicture}(0,-0.5)(3,2.5) \pspolygon(0,0)(3,0)(2,2) \psline(2,0)(2,2) \uput[u](2,2){$A$} \uput[d](0,0){$B$} \uput[d](3,0){$C$} \uput[d](2,0){$G$} \end{pspicture} \qquad\qquad \begin{pspicture}(0,-0.5)(3,2.5) \pspolygon(0,0)(3,0)(1.7,2) \psline(1.7,0)(1.7,2) \psline(0,0)(2.3,2)(3,0) \uput[u](2.3,2){$D$} \uput[u](1.7,2){$K$} \uput[d](0,0){$E$} \uput[d](3,0){$F$} \uput[d](1.7,0){$H$} \end{pspicture} \end{center} \begin{cozum}\mbox{} \begin{compactenum} \item $AGB=KHE$ [4.\ post\"ulat] \item $KE=AB$ [4.\ \"onerme] \item $AGC$ ile $KHF$, dik a\c c\i lar [13.\ \"onerme] \item $AGC=KHF$ [4.\ post\"ulat] \item $GC=HF$ [3.\ genel kavram] \item $AC=KF$ [4.\ \"onerme] \item $KE=DE$ ve $KF=DF$ [1.\ genel kavram] \item $K$ ile $D$, farkl\i\ olamaz [7.\ \"onerme] \end{compactenum} \end{cozum} \textbf{Soru 3.} \mbox{} \begin{compactitem} \item A\c c\i lar\i n\i n biri dik olan, \item en k\i sa kenar\i\ verilmi\c s do\u gru olan, \item o kenar\i\ g\"oren a\c c\i s\i\ kalan dik olmayan a\c c\i s\i n\i n yar\i s\i\ olan \end{compactitem} bir \"u\c cgen in\c sa edin. (Yani en k\i sa kenar\i\ verilmi\c s bir do\u gru olan bir `30-60-90' \"u\c cgen in\c sa edin.) \begin{cozum} \mbox{} \begin{compactenum} \item Verilmi\c s do\u gru $AB$ olsun \item Bu do\u gru, $C$ noktas\i na uzat\i ls\i n [2. post\"ulat] \item $BC=AB$ olsun [3.\ \"onerme] \item $AC$ do\u grusunda $ACD$ e\c skenar \"u\c cgen in\c sa edilsin. [1.\ \"onerme] \item $DB$ birle\c stirilsin [1. post\"ulat] \item $ADB=BDC$ [8.\ \"onerme] \item $DAB=ADC$ [5.\ \"onerme] \item $ADB$, $DAB$ a\c c\i s\i n\i n yar\i s\i \item $ADB$ \"u\c cgeni, istenen \"u\c cgendir \end{compactenum} \end{cozum} \textbf{Soru 4.} Aşağıdaki önermede hangi ad\i m, \"Oklid'in ilk 4 post\"ulat\i\ ve ilk 20 \"onermesi kullan\i larak do\u grulanamaz. \begin{compactenum} \item T\"um e\c skenar \"u\c cgenlerin t\"um a\c c\i lar\i n\i n birbirine e\c sit oldu\u gunu g\"osterece\u giz. \item $ABC$ ile $DEF$, e\c skenar \"u\c cgenler olsun. \item $A$ a\c c\i s\i n\i n $D$ a\c c\i s\i na e\c sit oldu\u gunu g\"osterece\u giz. \item $AB=DE$ ise, $BC=DF$ ve $AC=DF$ de olur. \item O halde $A$ a\c c\i s\i\ $D$ a\c c\i s\i na e\c sit olmal\i. \item $AB