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diff --git a/mat/domace_naloge/8/dokument.tex b/mat/domace_naloge/8/dokument.tex new file mode 100644 index 0000000..a179ea9 --- /dev/null +++ b/mat/domace_naloge/8/dokument.tex @@ -0,0 +1,378 @@ +% !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}{1} +\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}{8} +\newcommand{\snovdn}{Toge preslikave } +\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{30. 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 5. oktobra 2020, in njihove + rešitve, ki sem jih spisal sam. Naloge obsegajo snov \textit{\snovdn}in so + iz učbenika (stran 26) in vaj (stran 21). Kjer je bilo potrebno izbrati neke poljubne primere, sem jih vedno izbral naključno. +\end{abstract} +% \tableofcontents + +\section{Učbenik \textit{Matematika 2} --- stran 26} +\begin{enumerate}[label=\textbf{\arabic*.}] + \item Prezrcali enakostranični trikotnik $ABC$, ki ima stranico dolgo \cm{1}, \textsuperscript{(kako lahko narišem razviden \cm{1} enakostranični trikotnik?!)} + \begin{enumerate}[label=\textbf{\xslalph*)}] % c, b + \begin{multicols}{2} + \setcounter{enumii}{1} + \item \color{cyan} čez nosilko stranice $BC$, \color{black} + \item \color{magenta} čez nosilko višine na osnovnico. \color{black} + \end{multicols} + \end{enumerate} + \begin{center} + \begin{tikzpicture} + \tkzDefPoint(0,0){A} + \tkzDefPoint(60:1){C} + \tkzDefPoint(1,0){B} + \tkzDrawLine(A,C) \tkzLabelLine[](A,C) + {$b=\color{magenta}a'\color{black}$} + \tkzDrawLine(C,B) \tkzLabelLine[pos = 0.25](C,B) + {$a=\color{magenta}b'\color{black}$} + \tkzDrawLine(B,A) \tkzLabelLine[below](B,A) + {$c=\color{magenta}c'\color{black}$} + \tkzDefMidPoint(C,B) \tkzGetPoint{CB} + \tkzCompass[color=cyan](C,CB) \tkzCompass[color=cyan](B,CB) + \tkzInterLC(A,CB)(CB,A) \tkzGetPoints{AAGAIN}{AX} + \tkzDrawLine[color=cyan](A,AX) + \tkzDrawPoint[color=cyan](AX) + \tkzLabelPoint[color=cyan](AX){$A'$} + \tkzDrawLine[color=cyan, add = 0 and 0](C,AX) + \tkzLabelLine[color=cyan](C,AX){$b'$} + \tkzDrawLine[color=cyan, add = 0 and 0](B,AX) + \tkzLabelLine[color=cyan](B,AX){$c'$} + \tkzDrawLine[color=cyan, add = 0 and 0](B,C) + \tkzLabelLine[color=cyan, pos = 0.75](C,B){$a'$} + \tkzDefMidPoint(A,B) \tkzGetPoint{AB} + \tkzCompass[color=magenta](A,AB) \tkzCompass[color=magenta](B,AB) + \tkzDrawLine[color=magenta](C,AB) + \tkzLabelPoint[below left](A){$A=\color{magenta}B'\color{black}$} + \tkzLabelPoint[](B) + {$B=\color{cyan}B'\color{black}=\color{magenta}A'\color{black}$} + \tkzLabelPoint[above](C) + {$C=\color{cyan}C'\color{black}=\color{magenta}C'\color{black}$} + \tkzDrawPoints(A,B,C) + \end{tikzpicture} + \end{center} + \item Načrtaj krožnico s središčem $S$ in polmerom \cm{3} ter premico $p$, ki je od točke $S$ oddaljena \cm{2}, ter prezrcali krožnico čez premico. + \begin{center} + \begin{tikzpicture} + \tkzDefPoint(0,0){S} + \tkzDefPoint(4,0){K} + \tkzCompass(S,K) + \tkzDefPoint(3,0){R} + \tkzDrawCircle(S,R) + \tkzDrawLine(S,K) + \tkzInterCC(S,K)(K,S) \tkzGetPoints{SK}{KS} + \tkzCompass(K,SK) \tkzCompass(K,KS) \tkzCompass(S,SK) \tkzCompass(S,KS) + \tkzDrawLine(SK,KS) \tkzLabelLine[left](SK,KS){$p$} + \tkzInterLL(SK,KS)(S,K) \tkzGetPoint{P} + \tkzMarkRightAngles(S,P,SK) + \tkzInterLC(S,K)(P,K) \tkzGetPoints{SAGAIN}{SX} \tkzCompass(P,SX) + \tkzInterLC(S,R)(P,R) \tkzGetPoints{RX}{RAGAIN} \tkzCompass(P,RX) + \tkzLabelPoint(SX){$S'$} + \tkzLabelPoint(RX){$R'$} + \tkzDrawCircle(SX,RX) + \tkzDrawPoints(S,R) + \tkzLabelPoints(S,R) + \end{tikzpicture} + \end{center} + \item Premici $p$ in $q$ se sekata v točki $T$. Prezrcali premico $p$ čez premico $q$, če premici oklepata kot: + \begin{enumerate}[label=\textbf{\xslalph*)}] % b, č + \begin{multicols}{2} + \setcounter{enumii}{1} + \item \color{cyan}\ang{60}\color{black} + \setcounter{enumii}{3} + \item \color{magenta}\ang{75}\color{black} + \end{multicols} + \end{enumerate} + \begin{center} + \begin{tikzpicture} + \tkzDefPoint(-4,0){P} + \tkzDefPoint(4,0){R} + \tkzDefPoint(30:4){Q} + \tkzDefPoint(210:4){W} + \tkzDefPoint(60:4){Q1} + \tkzDefPoint(240:4){W1} + \tkzDefMidPoint(P,R) \tkzGetPoint{T} + \tkzDrawLine(P,R) \tkzLabelLine[above, pos = 0.25](P,R){$p$} + \tkzDrawLine[color=cyan](Q,W) \tkzLabelLine[above, pos = 0.25, color=cyan](Q,W){$q$} % cayn + \tkzCompass[color=cyan](T,W) \tkzInterCC(T,W)(W,P) \tkzGetPoints{PX}{PAGAIN} + \tkzCompass[color=cyan](W,PX) \tkzCompass[color=cyan](T,PX) \tkzLabelPoint[color=cyan](PX){$P'$} + \tkzCompass[color=cyan](T,Q) \tkzInterCC(T,Q)(Q,R) \tkzGetPoints{RX}{RAGAIN} + \tkzCompass[color=cyan](Q,RX) \tkzCompass[color=cyan](T,RX) \tkzLabelPoint[color=cyan](RX){$R'$} + \tkzDrawLine[color=cyan](RX,PX) + \tkzDrawLine[color=magenta](Q1,W1) \tkzLabelLine[color=magenta, above, pos = 0.25](Q1,W1){$q$} % magen + \tkzCompass[color=magenta](T,W1) \tkzInterCC(T,W1)(W1,P) \tkzGetPoints{PX1}{PAGAIN1} + \tkzCompass[color=magenta](W1,PX1) \tkzCompass[color=magenta](T,PX1) \tkzLabelPoint[color=magenta](PX1){$P'$} + \tkzCompass[color=magenta](T,Q1) \tkzInterCC(T,Q1)(Q1,R) \tkzGetPoints{RX1}{RAGAIN1} + \tkzCompass[color=magenta](Q1,RX1) \tkzCompass[color=magenta](T,RX1) \tkzLabelPoint[color=magenta](RX1){$R'$} + \tkzDrawLine[color=magenta](RX1,PX1) + \tkzDrawPoints(P,R,T) + \tkzLabelPoints(P,R,T) + \end{tikzpicture} + \end{center} + \item Zavrti enakostranični trikotnik $ABC$ okrog oglišča $C$ za kot: + \begin{enumerate}[label=\textbf{\xslalph*)}] % č, b + \begin{multicols}{2} + \setcounter{enumii}{1} + \item \color{cyan} \ang{60} \color{black} + \setcounter{enumii}{3} + \item \color{magenta} \ang{120} \color{black} + \end{multicols} + \begin{center} + \begin{tikzpicture} + \tkzDefPoint(0,0){A} + \tkzDefPoint(4,0){B} + \tkzDefPoint(60:4){C} + \tkzDrawLine(A,B) \tkzLabelLine[above](A,B){$c$} + \tkzDrawLine(A,C) \tkzLabelLine[below](A,C){$b$} + \tkzDrawLine(C,B) \tkzLabelLine[below](C,B){$a=\color{cyan}b'$} + \tkzInterCC(C,B)(B,C) \tkzGetPoints{CBU}{CBUNAPAK} + \tkzCompass[color=cyan](C,CBU) \tkzCompass[color=cyan](B,CBU) + \tkzDrawLine[color=cyan](C,B) + \tkzDrawLine[color=cyan](C,CBU) + \tkzLabelLine[above](C,CBU) + {$\color{cyan}a'\color{black}=\color{magenta}b'$} + \tkzDrawLine[color=cyan](B,CBU) + \tkzLabelLine[color=cyan, above](B,CBU){$c'$} + \tkzDrawLine[color=magenta](C,CBU) + \tkzInterLC(A,C)(C,A) \tkzGetPoints{AAGAIn}{AX} + \tkzDrawLine[color=magenta](C,AX) \tkzCompass[color=magenta](C,AX) + \tkzLabelLine[above, color=magenta](C,AX){$a'$} + \tkzDrawLine[color=magenta](AX,CBU) + \tkzLabelLine[color=magenta, above](AX,CBU){$c'$} + \tkzLabelPoint[color=magenta](AX){$A'$} + \tkzLabelPoint(CBU){$\color{cyan}B'\color{black}=\color{magenta}A'$} + \tkzLabelPoint(B){$B=\color{cyan}A'$} + \tkzLabelPoint(C){$C=\color{cyan}C'\color{black}=\color{magenta}C'$} + \tkzDrawPoints(A,B,C) \tkzLabelPoints(A) + \end{tikzpicture} + \end{center} + \end{enumerate} + \item Zavrti okrog središča: + \begin{enumerate}[label=\textbf{\xslalph*)}] % c, a + \begin{multicols}{2} + \item kvadrat za \ang{90} + \begin{center} + \begin{tikzpicture} + \tkzDefPoint(0,1){D} \tkzDefPoint(1,1){C} + \tkzDefPoint(0,0){A} \tkzDefPoint(1,0){B} + \tkzDrawLine(A,B) \tkzDrawLine(B,C) + \tkzDrawLine(C,D) \tkzDrawLine(D,A) + \tkzDrawPoints(A,B,C,D) + \tkzLabelPoint[below left](A){$A=D'$} + \tkzLabelPoint[](B){$B=A'$} + \tkzLabelPoint[above right](C){$C=B'$} + \tkzLabelPoint[above left](D){$D=C'$} + \end{tikzpicture} + \end{center} + \setcounter{enumii}{2} + \item krožnico za \ang{90} + \begin{center} + \begin{tikzpicture} + \tkzDefPoint(0,0){S} + \tkzDefPoint(1,0){K} + \tkzDrawCircle(S,K) + \tkzDrawPoints(S) + \tkzLabelPoints(S) + \end{tikzpicture} + \end{center} + \end{multicols} + \end{enumerate} + \item Prezrcali + \begin{enumerate}[label=\textbf{\xslalph*)}] % c, b + \setcounter{enumii}{1} + \item pravilni šestkotnik $ABCDEF$ čez eno izmed oglišč, + \item krožnico čez eno izmed točk na njej. + \begin{multicols}{2} + \begin{center} + \begin{tikzpicture} + \tkzDefPoint(0,0){O} + % šestkotnik 360/6=60 + \foreach[count=\i] \ANG in {60,60*2,60*3,60*4,60*5,60*6} { + \tkzDefPoint({2*cos(\ANG*pi/180)},{2*sin(\ANG*pi/180)}){P\i} + } + \foreach[count=\i] \ANG in {60,60*2,60*3,60*4,60*5,60*6} { + \tkzDefPoint({2*cos(\ANG*pi/180)+4},{2*sin(\ANG*pi/180)}){R\i} + } + \tkzDrawPolygon[](P1,P2,P3,P4,P5,P6) + \tkzDrawPoints(P1,P2,P3,P4,P5,P6) + \tkzDrawPolygon[color=magenta](R1,R2,R3,R4,R5,R6) + \tkzDrawPoints[color=magenta](R1,R2,R3,R4,R5,R6) + \tkzDrawLine[color=cyan](P2,R5) \tkzDrawLine[color=cyan](P4,R1) + \tkzDrawLine[add=0.1 and 0.1, color=cyan](P3,R6) + \tkzCompass[color=cyan](P6,R5) \tkzCompass[color=cyan](P6,R6) + \tkzCompass[color=cyan](P6,R1) \tkzCompass[color=cyan](P6,R2) + \tkzCompass[color=cyan](P6,R4) + \end{tikzpicture} + \end{center} + \begin{center} + \begin{tikzpicture} + \tkzDefPoint(0,0){S} + \tkzDefPoint(1.5,0){K} + \tkzDefPoint(3,0){SX} + \tkzDrawCircle(S,K) + \tkzDrawCircle[color=magenta](SX,K) + \tkzDrawLine[color=cyan](S,SX) + \tkzCompass[color=cyan](K,SX) + \tkzLabelPoint[color=magenta](SX){$S'$} + \tkzDrawPoints(S) \tkzLabelPoints(S) + \tkzDrawPoints[color=magenta](K) + \end{tikzpicture} + \end{center} + \end{multicols} + \end{enumerate} +\end{enumerate} +\section{Zbirka nalog \textit{Matematika 2} --- stran 21} +\begin{enumerate}[label=\textbf{\arabic*.}] % 113, 108, 112, 105 + \setcounter{enumi}{104} + \item Slike preriši na karirast papir v zvezek. Nariši zrcalne slike likov glede na narisano premico. + \setcounter{enumi}{107} + \item V pravokotnem sistemu nariši daljivo s krajiščema $A(3,-2)$ in $B(6,1)$. Nariši daljice: + \begin{itemize} + \item $A_1B_1$, ki je zrcalna slika daljice $AB$ glede na os $x$, + \item $A_2B_2$, ki je zrcalna slika daljice $AB$ glede na os $y$, + \item $A_3B_3$, ki je zrcalna slika daljice $AB$ glede na simetralo lihih kvadrantov, + \item $A_4B_4$, ki je zrcalna slika daljice $AB$ glede na simetralo sodih kvadrantov, + \end{itemize} + in zapiši koordinate krajišč teh daljic. + \setcounter{enumi}{111} + \item Nariši: + \begin{enumerate}[label=\textbf{\xslalph*)}] + \item trikotnik s podatki $a=\cm{5}$, $b=\cm{4}$ in $c=\cm{4}$ ter ga prezrcali čez stranico $c$, + \item enakostranični trikotnik $ABC$ s \cm{3} dolgo stranico ter ga prezrcali čez stranico $AC$, + \item kvadrat $ABCD$ s stranico, dolgo \cm{2}, in ga prezrcali čez stranico $AD$, + \item pravokotnik $ABCD$ s stranicama, dolgima $a=\cm{4}$ in $b=\cm{2}$, in ga prezrcali čez diagonalo $AC$. + \end{enumerate} + \item Slike preriši na karirast papir. Nariši vse simetrale. +\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} |