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diff --git a/mat/domace_naloge/6/dokument.tex b/mat/domace_naloge/6/dokument.tex new file mode 100644 index 0000000..51b6a5d --- /dev/null +++ b/mat/domace_naloge/6/dokument.tex @@ -0,0 +1,315 @@ +% !TeX encoding = UTF-8 +% !TeX spellcheck = sl_SI +\documentclass[]{article} +\usepackage[utf8]{inputenc} +\usepackage{etoolbox} +\usepackage[hidelinks]{hyperref} +\usepackage[a4paper, total={7in, 10in}]{geometry} +\usepackage{graphicx} +\usepackage{hologo} +\usepackage{amssymb} +\usepackage{pgfplots} +\usepackage[inline]{enumitem} +\usepackage{amsmath} +\usepackage{siunitx} +\usepackage{multicol} +\usepackage{tabularcalc} +\usepackage{amsmath} +\usepackage{float} +\usepackage{tasks} +\usepackage{filecontents} +\usepackage{textcomp} +\usepackage{tkz-euclide} +\usepackage{listings} +% \usepackage{multienum} % weird with labels +\usetikzlibrary{calc} %% not really needed............. idk. +%\usetikzlibrary{external} +% \usetkzobj{all} % tkz-euclide > 3.02 tega ne potrebuje več (: +%\tikzexternalize +\usepackage{xcolor} +\sisetup{output-decimal-marker = {,}, quotient-mode=fraction,per-mode=fraction} % per-mode=symbol +\newcommand\ddfrac[2]{\frac{\displaystyle #1}{\displaystyle #2}} +\newcommand{\functionSamples}{100} % fix to fancier value upon release, keep low during development +\newcommand{\razhroscevanje}{0} +\definecolor{codegreen}{rgb}{0,0.6,0} +\definecolor{codegray}{rgb}{0.5,0.5,0.5} +\definecolor{codepurple}{rgb}{0.58,0,0.82} +\definecolor{backcolour}{rgb}{0.95,0.95,0.92} + +\lstdefinestyle{mystyle}{ + backgroundcolor=\color{backcolour}, + commentstyle=\color{codegreen}, + keywordstyle=\color{magenta}, + numberstyle=\tiny\color{codegray}, + stringstyle=\color{codepurple}, + basicstyle=\ttfamily\footnotesize, + breakatwhitespace=false, + breaklines=true, + captionpos=b, + keepspaces=true, + numbers=left, + numbersep=5pt, + showspaces=false, + showstringspaces=false, + showtabs=false, + tabsize=2 +} + +\lstset{style=mystyle} +\def\@maketitle{% + \newpage + \null + \vskip 2em% + \begin{center}% + \let \footnote \thanks + {\LARGE \@title \par}% + \vskip 1.5em% + {\large + \lineskip .5em% + \begin{tabular}[t]{c}% <------ + \@author% <------ Authors + \end{tabular}\par}% <------ + \vskip 1em% + {\large \@date}% + \end{center}% + \par + \vskip 1.5em} +%opening +\makeatletter +\newcommand{\stevilkadn}{6} +\newcommand{\snovdn}{Skladnost trikotnikov } +\newcommand{\cm}[1]{\SI{#1}{\centi\meter}} +\newcommand{\xslalph}[1]{\expandafter\@xslalph\csname c@#1\endcsname} +\newcommand{\@xslalph}[1]{% + \ifcase#1\or a\or b\or c\or \v{c}\or d\or e\or f\or g\or h\or i% + \or j\or k\or l\or m\or n\or o\or p\or r\or s\or \v{s}% + \or t\or u\or v\or z\or \v{z} + \else\@ctrerr\fi% +} +\AddEnumerateCounter{\xslalph}{\@xslalph}{m} +\makeatother +\newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma +\newcommand*\textfrac[2]{ + \frac{\text{#1}}{\text{#2}} +} +\renewcommand\abstractname{Povzetek} +\date{17. september 2020} +\renewcommand\contentsname{Kazalo vsebine} +\renewcommand\figurename{Slika} +\renewcommand\abstractname{Povzetek} +\newcommand{\iic}{I\textsuperscript{2}C } +% \patchcmd{\thebibliography}{\section*{\refname}}{}{}{} +\title{% + \snovdn --- \stevilkadn. domača naloga \\ + \large Matematika, Gimnazija Bežigrad} +\author{\begin{tabular}{rl} + \textbf{Profesor:} & prof. Vilko Domajnko \\ + \textbf{Avtor:} & Anton Luka Šijanec, 2. a +% \textbf{Avtor:} & Anton Luka Šijanec \\ & Member 2 \\ & Member 3 +\end{tabular}} + + +% \everymath{\displaystyle} % https://tex.stackexchange.com/a/32847/212260 +\begin{document} +\maketitle + +\begin{abstract} + Ta dokument obsega naloge, naročene dijakom 16. septembra 2020, in njihove + rešitve, ki sem jih spisal sam. Naloge obsegajo snov \textit{\snovdn}in so + iz učbenika in vaj/zbirke nalog. +\end{abstract} +% \tableofcontents + +\section{Učbenik \textit{Matematika 1}: Stran 19} +\begin{enumerate}[label=\textbf{\arabic*.}] + \item Načrtaj kote, velike \ang[parse-numbers=false]{\textbf{105}}, \ang{150}, \ang{15}, \ang{135} in \ang{165}. + \begin{center} + \begin{tikzpicture} + \tkzDefPoint(-5,0){A} %%% kot 90 stopinj ˇˇˇ + \tkzDefPoint(5,0){B} + \tkzDefPoint(0,0){CHEATING0} \tkzDefPoint(0.05,0){CHEATING1} \tkzDefPoint(-0.05,0){CHEATING2} + \tkzInterCC(A,B)(B,A) \tkzGetPoints{C}{D} + \tkzDrawPoints(A) \tkzCompass(A,C) \tkzCompass(B,C) \tkzCompass(A,CHEATING1) \tkzCompass(B,CHEATING2) \tkzCompass(A,B) + \tkzDrawLine(A,B) \tkzDrawLine(CHEATING0,C) \tkzMarkAngle[mark=none](B,CHEATING0,C) + \tkzMarkRightAngles(B,CHEATING0,C) + %%% kot 60 stopinj ˇˇˇ + \tkzInterLC(CHEATING0,C)(CHEATING0,A) \tkzGetPoints{E}{F} \tkzCompass(CHEATING0,F) + \tkzInterCC(CHEATING0,F)(F,CHEATING0) \tkzGetPoints{G}{H} \tkzCompass(CHEATING0,G) \tkzCompass(F,G) + \tkzDrawLine(CHEATING0,G) \tkzMarkAngle[color=magenta, mark=none](F,CHEATING0,G) + %%% kot 60/2 stopinj ˇˇˇ + \tkzDrawLine(F,G) \tkzDefMidPoint(F,G) \tkzGetPoint{I} \tkzCompass(F,I) \tkzCompass(G,I) + \tkzMarkAngle[color=cyan,mark=none, size=1.1cm](F,CHEATING0,I) + %%% kot 60/2/2 stopinj = 15 stopinj ˇˇˇ + \tkzDefMidPoint(F,I) \tkzGetPoint{J} \tkzCompass(F,J) \tkzCompass(I,J) + \tkzMarkAngle[color=yellow,mark=none, size=1.2cm](F,CHEATING0,J) + %%% kot 105 stopinj = kot 90 stopinj + kot 15 stopinj ˇˇˇ + \tkzMarkAngle[color=red,mark=none, size=0.9cm](B,CHEATING0,J) \tkzDrawLine(CHEATING0,J) + \end{tikzpicture} + \end{center} + \item Načrtaj trikotnik, ki ima stranice dolge \SI{6}{\centi\meter}, \SI{4}{\centi\meter} in \SI{4,5}{\centi\meter}. + + \item Načrtaj trikotnik z danimi podatki. + \begin{multicols}{4} + \noindent + \begin{enumerate}[label=\textbf{\xslalph*)}] + \item $\mathbf{c=}\SI[parse-numbers=false, parse-units=false]{\mathbf{5}}{\mathbf{\centi\meter}}$ + + $\mathbf{b=}\SI[parse-numbers=false, parse-units=false]{\mathbf{4}}{\mathbf{\centi\meter}}$ + + $\mathbf{\alpha=}\ang[parse-numbers=false, parse-units=false]{\mathbf{75}}$ + \item $a=\SI{6}{\centi\meter}$ + + $\beta=\ang{60}$ + + $\gamma=\ang{45}$ + \item $c=\SI{5,5}{\centi\meter}$ + + $a=\SI{4}{\centi\meter}$ + + $\gamma=\ang{75}$ + \item $b=\SI{5}{\cm}$ + + $c=\SI{6,5}{\centi\meter}$ + + $\beta=\ang{60}$ + \end{enumerate} + \end{multicols} + \begin{multicols}{2} + \noindent + \begin{center} + \begin{tikzpicture} + \tkzDefPoint(0,0){A} + \tkzDefPoint(5,0){B} + \tkzDefPoint(4,0){b} + \tkzDrawPoints(A) \tkzCompass(A,b) \tkzDrawLine(A,B) \tkzCompass(A,b) + \tkzDrawSegment[dim={$\SI{4,0}{\centi\meter}$,6pt,transform shape}](A,b) + \tkzInterCC(A,b)(b,A) \tkzGetPoints{D}{d} \tkzCompass(A,D) \tkzCompass(B,D) \tkzInterCC(D,A)(A,D) \tkzGetPoints{e}{E} + \tkzCompass(A,E) \tkzCompass(D,E) \tkzDefMidPoint(E,D) \tkzGetPoint{F} \tkzCompass(D,F) \tkzCompass(E,F) + \tkzDefMidPoint(F,D) \tkzGetPoint{C} \tkzCompass(F,C) \tkzCompass(D,C) \tkzDrawLine(A,C) \tkzDrawLine(B,C) + \tkzLabelLine[below left](B,C){$a$} \tkzLabelLine[below right](C,A){$b$} \tkzLabelLine[below](A,B){$c$} + \tkzLabelPoints(A) \tkzLabelPoints[below left](B) \tkzLabelPoints[above right](C) + \tkzMarkAngle[mark=none](B,A,C) \tkzLabelAngle[pos=1.2](B,A,C){$\alpha$} + \end{tikzpicture} + \begin{tikzpicture} + \tkzDefPoint(0,0){A} + \tkzDefPoint(6,0){B} + \tkzDefPoint(4,0){a} + \tkzDefPoint(4.5+6,0){b} + \tkzDrawPoint(A) + \tkzCompass(A,B) + \tkzInterCC(A,a)(B,b) \tkzGetPoints{C}{c} \tkzCompass(A,C) \tkzCompass(B,C) + \tkzDrawLine(A,B) \tkzDrawLine(B,C) \tkzDrawLine(C,A) + \end{tikzpicture} + \end{center} + \end{multicols} + \item Načrtaj štirikotnik z danimi podatki. Pomagaj si s trikotnika, na katera diagonala razdeli štirikotnik + \begin{multicols}{6} + \noindent + \begin{enumerate}[label=\textbf{\xslalph*)}] + \item $\mathbf{a=}\SI[parse-numbers=false, parse-units=false]{\mathbf{5}}{\mathbf{\centi\meter}}$ + + $\mathbf{b=}\SI[parse-numbers=false, parse-units=false]{\mathbf{3}}{\mathbf{\centi\meter}}$ + + $\mathbf{c=}\SI[parse-numbers=false, parse-units=false]{\mathbf{4}}{\mathbf{\centi\meter}}$ + + $\mathbf{d=}\SI[parse-numbers=false, parse-units=false]{\mathbf{4}}{\mathbf{\centi\meter}}$ + + $\mathbf{f=}\SI[parse-numbers=false, parse-units=false]{\mathbf{5}}{\mathbf{\centi\meter}}$ + \item $a=\SI{6}{\centi\meter}$ + + $b=\SI{5}{\centi\meter}$ + + $d=\SI{4}{\centi\meter}$ + + $\alpha=\ang{120}$ + + $\delta=\ang{90}$ + \item $b=\SI{6}{\centi\meter}$ + + $c=\cm{2}$ + + $d=\cm{5}$ + + $\gamma=\ang{105}$ + + $\delta=\ang{75}$ + \item $a=\cm{7}$ + + $b=\cm{4}$ + + $c=\cm{5}$ + + $e=\cm{6}$ + + $f=\cm{5,5}$ + \item $d=c=\cm{3}$ + + $a=d=\cm{5}$ + + $f=\cm{5,5}$ + \item $a=\cm{5}$ + + $d=\cm{4}$ + + $e=\cm{6}$ + + $f=\cm{7}$ + + $\beta=\ang{60}$ + \end{enumerate} + \end{multicols} + % \begin{center} + Štirikotnikov še nismo definirali. (-; Problem je v tem, da ne vem, med katerimi pari točk so določene stranice. + % \end{center} +\end{enumerate} + +\section{Zbirka nalog \textit{Matematika 1}: Stran 16} +\begin{enumerate}[label=\textbf{\arabic*.}] + \setcounter{enumi}{71} + \item Trikotnika $ABC$ in $A'B'C$ sta skladna. Točki $D$ in $D'$ ležita na stranicah $AB$ oziroma $A'B'$ tako, da sta kota $DCB$ in $D'C'B'$ skladna. Dokaži, da je $|AD|=|A'D'|$ + \item Na sliki je $|AE|=|BG|$, kota $BAC$ in $FEG$ sta skladna, prav tako sta skladna kota $CBA$ in $EGF$. Dokaži, da sta trikotnika $ABC$ in $EGF$ skladna. + \item Točke $A, B, C, E$ in $F$ ležijo, kot kaže slika: $|AB|=|BC|, AF \bot BE, CE \bot BF$. Dokaži, da je $|AF|=|CE|$. + \begin{multicols}{2} + \begin{center} + \begin{tikzpicture} + \tkzDefPoint(0,0){B} + \tkzDefPoint(-5,0){C} + \tkzDefPoint(210:5){A} + %\tkzDefPoint(210:6){E} + \tkzDefLine[orthogonal=through A](B,A) \tkzGetPoint{f} + \tkzInterLL(A,f)(B,C) + \tkzGetPoint{F} + \tkzDefMidPoint(A,F) \tkzGetPoint{M} + \tkzDefLine[orthogonal=through M](B,F) \tkzGetPoint{e} + \tkzInterLL(C,e)(B,A) \tkzGetPoint{E} + \tkzDrawPoints(A,B,C,F,E) + \tkzLabelPoints(B,E,A) \tkzLabelPoints[above right](F,C) + \tkzDrawLine(B,F) \tkzDrawLine[add = 0 and 0](C,E) + \tkzDrawLine(B,E) \tkzDrawLine[add = 0 and 0](A,F) + \tkzMarkSegments[mark=|](C,B A,B) \tkzMarkSegments[mark=||](F,A E,C) + \tkzMarkAngles[mark=|||](C,B,A) \tkzLabelAngle[pos=2](C,B,A){$\beta$} + \tkzMarkRightAngles(B,A,F E,C,B) + \end{tikzpicture} + \end{center} + + Trikotnika $ABF$ in $EBC$ sta skladna zaradi dveh skladnih kotov in ene stranice, zato so vse njune stranice enako dolge. + \end{multicols} +\end{enumerate} + +\section{Zaključek} +Ta dokument je informativne narave in se lahko še spreminja. Najnovejša različica, torej PDFji in +\hologo{LaTeX} +izvorna koda, zgodovina sprememb in prejšnje različice, je na voljo mojem šolskem Git repozitoriju na +\url{https://github.com/sijanec/sola-gimb-2} v mapi +\href{https://github.com/sijanec/sola-gimb-2/tree/master/mat/domace_naloge/\stevilkadn/}{/mat/domace\_naloge/\stevilkadn/}. Povezava za ogled zadnje različice tega dokumenta v PDF obliki je \url{http://razor.arnes.si/~asija3/files/sola/gimb/2/mat/domace_naloge/\stevilkadn/dokument.pdf} in/ali \url{https://github.com/sijanec/sola-gimb-2/raw/master/mat/domace_naloge/\stevilkadn/dokument.pdf}. + +\if\razhroscevanje1 +\section{Razhroščevalne informacije} +Te informacije so generirane, ker je omogočeno razhroščevanje. Prej objavo dokumenta izklopite razhroščevanje. To naredite tako, da nastavite ukaz \texttt{razhroscevanje} na 0 v začetku dokumenta. + +Grafi imajo natančnost \functionSamples\space točk na graf. + +Konec generiranja dokumenta: \input|"date -Ins" +\fi +% \item $$$$ +\end{document} |