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authorsijanec <sijanecantonluka@gmail.com>2020-10-01 00:12:18 +0200
committersijanec <sijanecantonluka@gmail.com>2020-10-01 00:12:18 +0200
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treeaacf065597b2f91c97a30a7ff2d642eff108fab9
parentzj (diff)
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+% !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}{7}
+\newcommand{\snovdn}{Zrcaljenje in simetrija }
+\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 30. septembra 2020, in njihove
+ rešitve, ki sem jih spisal sam. Naloge obsegajo snov \textit{\snovdn}in so
+ iz učnega lista profesorja Domajnka.
+\end{abstract}
+% \tableofcontents
+
+\section{Učni list \textit{\snovdn} -- naloge}
+\begin{enumerate}[label=\textbf{\arabic*.}] % 2, 6, 3, 1
+ \item \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Nariši \color{magenta}enakokraki trikotnik \color{black} s podatki: $|AC|=|BC|=2|AB|=\SI{8}{\cm}$. Prezrcali ta lik čez:
+ \begin{enumerate}[label=\textbf{a\textsubscript{\arabic*)}}]
+ \item premico skozi točki $A$ in $B$,
+ \item premico skozi točki $A$ in $C$,
+ \item premico, ki gre skozi točko $C$ in razpolovišče daljice $AB$,
+ \item premico, ki gre skozi razpolovišči daljic $AC$ in $BC$,
+ \item \color{cyan} premico skozi točko $A$, ki je vzporedna premici skozi točki $B$ in $C$. \color{black}
+ \end{enumerate}
+ \item Naj bo $ABC$ trikotnik, za katerega velja: $a=\SI{5}{\cm}, b=\SI{6}{\cm}, c=\SI{7}{\cm}$.
+ \begin{enumerate}[label=\textbf{b\textsubscript{\arabic*)}}]
+ \item Načrtaj trikotnik $ABC$.
+ \item Načrtaj pravokotno projekcijo točke $C$ na daljico $AB$.
+ \item Trikotnik $ABC$ prezrcali čez nosilko stranice $BC$.
+ \item Naj bo $D$ razpolovišče daljice $AB$. Načrtaj točko $D$. Trikotnik $ABC$ prezrcali čez premico skozi $c$ in $D$.
+ \item Naj bo $E$ taka točka na daljici $AB$, da je $AB\bot CE$. Načrtaj točko $E$. Trikotnik $ABC$ rezrcali čez premico skozi $C$ in $E$.
+ \end{enumerate}
+ \end{enumerate}
+ \item \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \setcounter{enumii}{2}
+ \item Nariši premici $p$ in $q$, ki se sekata pod kotom \ang{30}. Prezrcali $p$ čez $q$.
+ \item \color{magenta} Nariši premici $p$ in $q$, ki sta vzporedni in oddaljeni \SI{5}{\cm}. Prezrcali $p$ čez $q$.\color{black}
+ \end{enumerate}
+ \item Naj bo \color{magenta}$AB$ \color{black} daljica. Poišči vse simetrale daljice \color{magenta}$AB$\color{black}.
+ \setcounter{enumi}{5}
+ \item \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Poišči vse simetrale krožnice s središčem v točki $S$ in s polmerom $r$. (\textbf{Krožnica} je množica točk, ki so vse enako oddaljene od središča.). Dokaži.
+ \item Naj bo $ABCD$ romb. (\textbf{Romb} je štirikotnik, ki ima vse stranice enako dolge.). Poišči vse simetrale romba $ABCD$. Dokaži. Ali je romb simetričen lik?
+ \item Naj bo $ABCD$ deltoid. (\textbf{Deltoid} je štirikotnik, ki ima po dve stranici enako dolgi.). Poišči vse simetrale deltoida $ABCD$. Dokaži.
+ \item Naj bo $ABCD$ kvadrat. (\textbf{Kvadrat} je štirikotnik, ki ima vse stranice enako dolge, poljuben par sosednjih stranic pa oklepa pravi kot). Poišči vse simetrale kvadrata $ABCD$. Dokaži.
+ \item Naj bo $ABCD$ pravokotnik, kjer je: $|AB|=2|BC|$. (\textbf{Pravokotnik} je štirikotnik, ki ima po dve nasprotni stranici enako dolgi, poljuben par sosednjih stranic pa oklepa pravi kot.). Poišči vse simetrale pravokotnika $ABCD$. Dokaži.
+ \end{enumerate}
+\end{enumerate}
+
+\section{Rešitve}
+\subsection{Nalogi 2 in 3}
+\begin{center}
+ \begin{tikzpicture} % naloga 2
+ \tkzDefPoint(-4,0){P}
+ \tkzDefPoint(4,0){R}
+ \tkzDefPoint(15:4){Q}
+ \tkzDefPoint(195:4){W}
+ \tkzDrawLine(P,R) \tkzLabelLine[pos=0.25, above](P,R){$p$}
+ \tkzDrawLine(Q,W) \tkzLabelLine[pos=0.75, below](Q,W){$q$}
+ \tkzDefPoint(-2.5,0){P2} % naredimo preslikavo za P
+ \tkzInterLC(Q,W)(P,P2) \tkzGetPoints{PP1}{PP2}
+ \tkzCompass(P,PP1) \tkzCompass(P,PP2)
+ \tkzInterCC(PP1,PP2)(PP2,PP1) \tkzGetPoints{PPZ1}{PPZ2}
+ \tkzCompass(PP1,PPZ1) \tkzCompass(PP2,PPZ1)
+ \tkzDrawLine(P,PPZ1)
+ \tkzInterLL(P,PPZ1)(Q,W) \tkzGetPoint{PS}
+ \tkzInterLC(P,PPZ1)(PS,P) \tkzGetPoints{PP}{PPX}
+ \tkzCompass(PS,PPX) \tkzLabelPoint(PPX){$P'$}
+ \tkzMarkRightAngle(P,PS,PP2)
+ \tkzDefPoint(2.5,0){R2} % pa še za R
+ \tkzInterLC(Q,W)(R,R2) \tkzGetPoints{RP1}{RP2}
+ \tkzCompass(R,RP1) \tkzCompass(R,RP2)
+ \tkzInterCC(RP1,RP2)(RP2,RP1) \tkzGetPoints{RPZ1}{RPZ2}
+ \tkzCompass(RP1,RPZ1) \tkzCompass(RP2,RPZ1)
+ \tkzDrawLine(R,RPZ1)
+ \tkzInterLL(R,RPZ1)(Q,W) \tkzGetPoint{RS}
+ \tkzInterLC(R,RPZ1)(RS,R) \tkzGetPoints{RP}{RPX}
+ \tkzCompass(RS,RPX) \tkzLabelPoint(RPX){$R'$}
+ \tkzDrawPoints(P,R,PPX,RPX)
+ \tkzDrawLine(PPX,RPX) \tkzLabelLine[pos=0.75, below](PPX,RPX){$p'$}
+ \tkzLabelPoints(P,R)
+ \tkzMarkRightAngle(R,RS,RP2)
+ % pa naredimo tukaj še b primer ... z magento
+ \tkzDefPoint(-4,5){BQ} \tkzDefPoint(4,5){BW}
+ \tkzDrawLine[color=magenta](BQ,BW)
+ \tkzLabelLine[below, color=magenta](BQ,BW){$q$}
+ \tkzDrawPoints[color=magenta](BQ,BW)
+ \tkzLabelPoint[color=magenta](BQ){$Q$}
+ \tkzLabelPoint[color=magenta](BW){$W$}
+ \tkzInterLC(P,R)(BQ,P2) \tkzGetPoints{BPQC1}{BPQC2} % naredimo preslik. za Q
+ \tkzCompass[color=magenta](BQ,BPQC1) \tkzCompass[color=magenta](BQ,BPQC2)
+ \tkzInterCC(BPQC1,BPQC2)(BPQC2,BPQC1) \tkzGetPoints{BPP1}{BPP2}
+ \tkzCompass[color=magenta](BPQC1,BPP2)
+ \tkzCompass[color=magenta](BPQC2,BPP2)
+ \tkzInterLC(BQ,BPP2)(P,BQ) \tkzGetPoints{BQP}{BQPX}
+ \tkzCompass[color=magenta](P,BQPX)
+ \tkzDrawPoints[color=magenta](BQPX)
+ \tkzLabelPoint[color=magenta](BQPX){$Q'=A$}
+ \tkzDrawLine[color=magenta](BQ,BQPX)
+ \tkzInterLC(P,R)(BW,R2) \tkzGetPoints{BPWC1}{BPWC2} % pa še za W
+ \tkzCompass[color=magenta](BW,BPWC1) \tkzCompass[color=magenta](BW,BPWC2)
+ \tkzInterCC(BPWC1,BPWC2)(BPWC2,BPWC1) \tkzGetPoints{BRP1}{BRP2}
+ \tkzCompass[color=magenta](BPWC1,BRP1)
+ \tkzCompass[color=magenta](BPWC2,BRP1)
+ \tkzInterLC(BW,BRP1)(R,BW) \tkzGetPoints{BWP}{BWPX}
+ \tkzCompass[color=magenta](R,BWPX)
+ \tkzDrawPoints[color=magenta](BWPX)
+ \tkzLabelPoint[color=magenta](BWPX){$W'=B$}
+ \tkzDrawLine[color=magenta](BW,BWPX)
+ \tkzDrawLine[color=magenta](BQPX,BWPX)
+ \tkzLabelLine[color=magenta, above, pos=0.40](BQPX,BWPX){$q'=AB$}
+ % pa naredimo tukaj še tretjo nalogo
+ \tkzDefMidPoint(BQPX,BWPX) \tkzGetPoint{3M}
+ \tkzCompass(BQPX,3M) \tkzCompass(BWPX,3M)
+ \tkzDefMidPoint(BQPX,3M) \tkzGetPoint{3QM}
+ \tkzDefMidPoint(BWPX,3M) \tkzGetPoint{3WM}
+ \tkzCompass(3M,3WM) \tkzCompass(3M,3QM)
+ \tkzInterCC(3WM,3QM)(3QM,3WM) \tkzGetPoints{3S1}{3S2}
+ \tkzCompass(3WM,3S1) \tkzCompass(3QM,3S1)
+ \tkzCompass(3WM,3S2) \tkzCompass(3QM,3S2)
+ \tkzDrawLine(3S1,3S2) \tkzLabelLine[pos=0.25, left](3S1,3S2){simetrala}
+ \end{tikzpicture}
+\end{center}
+\newpage
+\subsection{Naloga 1}
+\begin{center}
+ \begin{tikzpicture}
+ \tkzDefPoint(-2,0){A} \tkzDefPoint(2,0){B}
+ \tkzDefPoint(-10,0){CCIRCLEHELPERA} \tkzDefPoint(10,0){CCIRCLEHELPERB}
+ \tkzCompass(A,B)
+ \tkzDrawLine[color=magenta](A,B)\tkzLabelLine[above,color=magenta](A,B){$c$}
+ \tkzInterCC(B,CCIRCLEHELPERB)(A,CCIRCLEHELPERA) \tkzGetPoints{CX}{C}
+ \tkzCompass(A,C) \tkzCompass(B,C) \tkzCompass(A,CX) \tkzCompass(B,CX)
+ \tkzDrawLine[color=magenta](A,C)
+ \tkzLabelLine[color=magenta,below right,pos=0.75](A,C){$b$}
+ \tkzDrawLine[color=magenta](B,C)
+ \tkzLabelLine[color=magenta,below left, pos=0.75](B,C){$a$}
+ \tkzDrawPoints[color=magenta](A,B,C)
+ \tkzLabelPoint(CX){$C'$} % prezrcali lik čez AB
+ \tkzDrawLine(A,CX) \tkzLabelLine[above right](A,CX){$b'$}
+ \tkzDrawLine(B,CX) \tkzLabelLine[above left](B,CX){$a'$}
+ \tkzLabelLine[below](A,B){$c'$}
+ \tkzInterLC(A,C)(B,A) \tkzGetPoints{ABUTAG}{BCIRC} \tkzCompass(B,BCIRC) % AC
+ \tkzInterCC(BCIRC,B)(A,B) \tkzGetPoints{BBUTAGAIN}{BX}
+ \tkzCompass(BCIRC,BX) \tkzCompass(A,BX) \tkzLabelPoint(CX){$C'$}
+ \tkzDrawLine(A,BX) \tkzLabelLine[above](A,BX){$c'$}
+ \tkzDrawLine(C,BX) \tkzLabelLine[below right](C,BX){$a'$}
+ \tkzLabelLine[left, pos=0.75](A,C){$b'$}
+ \tkzLabelLine[right, pos=0.75](B,C){$a'$} \tkzLabelPoint[right](BX){$B'$}
+ \tkzLabelPoint[color=magenta, above right](A){$A=A'$} % skozi C|AB
+ \tkzLabelPoint[color=magenta, above left](B){$B=B'$}
+ \tkzLabelPoint[color=magenta](C){$C=C'$}
+ \tkzDefMidPoint(A,C) \tkzGetPoint{ACM} \tkzCompass(A,ACM) \tkzCompass(C,ACM)
+ \tkzDefMidPoint(B,C) \tkzGetPoint{BCM} \tkzCompass(B,BCM) \tkzCompass(C,BCM)
+ \tkzDrawLine(ACM,BCM) \tkzInterCC(ACM,C)(BCM,C) \tkzGetPoints{CBUTAGAI}{CX2}
+ \tkzCompass(ACM,CX2) \tkzCompass(BCM,CX2)
+ \tkzLabelPoint(CX2){$C'$}
+ \tkzInterLC(BCM,ACM)(BCM,ACM) \tkzGetPoints{BCMOUTER}{BCMBUTAGAIN}
+ \tkzInterLC(BCM,ACM)(ACM,BCM) \tkzGetPoints{ABCMBUTAGAIN}{ACMOUTER}
+ \tkzCompass(ACM,ACMOUTER) \tkzCompass(BCM,BCMOUTER)
+ \tkzInterCC(BCMOUTER,B)(BCM,B) \tkzGetPoints{BBUTAGAI2}{BX2}
+ \tkzCompass(BCM,BX2) \tkzCompass(BCMOUTER,BX2)
+ \tkzLabelPoint[below left](BX2){$B'$}
+ \tkzInterCC(ACMOUTER,A)(ACM,A) \tkzGetPoints{AX}{ABUTAGAI2}
+ \tkzCompass(ACM,AX) \tkzCompass(ACMOUTER,AX)
+ \tkzLabelPoint[below right](AX){$A'$}
+ \tkzDrawLine(AX,BX2) \tkzLabelLine[pos=0.25](AX,BX2){$c'$}
+ \tkzDrawLine(BX2,CX2) \tkzLabelLine[](BX2,CX2){$a'$}
+ \tkzDrawLine(CX2,AX) \tkzLabelLine[](CX2,AX){$b'$}
+ \tkzCompass(B,BCMOUTER) \tkzCompass(A,ACMOUTER)
+ \tkzInterLC(B,C)(A,B) \tkzGetPoints{ACIRC}{BAGAIN} \tkzCompass(A,ACIRC)%cyan
+ \tkzDefMidPoint(B,ACIRC) \tkzGetPoint{BACIRCMID}
+ \tkzCompass[color=cyan](B,BACIRCMID)\tkzCompass[color=cyan](ACIRC,BACIRCMID)
+ \tkzDrawLine[color=cyan, add = 0.1 and 1.1](BACIRCMID,A)
+ \tkzMarkRightAngle[color=cyan](A,BACIRCMID,B)
+ \tkzInterLC(A,BACIRCMID)(A,BACIRCMID) \tkzGetPoints{BACIRCMIDP}{BACIRCMIDA}
+ \tkzCompass[color=cyan](A,BACIRCMIDP)
+ \tkzInterCC(BACIRCMID,BACIRCMIDP)(BACIRCMIDP,BACIRCMID)
+ \tkzGetPoints{BACIRCMID2P1}{BACIRCMID2P2}
+ \tkzCompass[color=cyan](BACIRCMIDP,BACIRCMID2P2)
+ \tkzCompass[color=cyan](BACIRCMID,BACIRCMID2P2)
+ \tkzDrawLine[color=cyan](BACIRCMID2P2,A)
+ \tkzInterLC(A,BACIRCMID2P2)(B,A) \tkzGetPoints{BCCIRC}{BAGAIN}
+ \tkzCompass[color=cyan](B,BCCIRC)
+ \tkzInterCC(A,B)(BCCIRC,B) \tkzGetPoints{ANOTHERB}{BP}
+ \tkzCompass[color=cyan](BCCIRC,BP) \tkzCompass[color=cyan](A,BP)
+ \tkzMarkRightAngle[color=cyan](BACIRCMID,A,BACIRCMID2P2)
+ \tkzInterLC(BACIRCMID2P2,A)(C,BACIRCMID2P2) \tkzGetPoints{CAGAIN}{CCIRC}
+ \tkzCompass[color=cyan](C,CCIRC)
+ \tkzInterCC(CCIRC,C)(BACIRCMID2P2,C) \tkzGetPoints{CAGAIN}{CP}
+ \tkzCompass[color=cyan](BACIRCMID2P2,CP) \tkzCompass[color=cyan](CCIRC,CP)
+ \tkzDrawPoints[color=cyan](CP,BP)
+ \tkzLabelPoint[color=cyan, right](CP){$C'$}
+ \tkzLabelPoint[color=cyan, above left](BP){$B'$}
+ \tkzDrawLine[color=cyan](BP,CP) \tkzLabelLine[color=cyan,right](BP,CP){$a'$}
+ \tkzDrawLine[color=cyan](BP,A) \tkzLabelLine[color=cyan, above](BP,A){$c'$}
+ \tkzDrawLine[color=cyan](A,CP) \tkzLabelLine[color=cyan, above](A,CP){$b'$}
+ \tkzDrawPoints(CX,BX,ACM,BCM,CX2,BX2,AX)
+ \end{tikzpicture}
+\end{center}
+
+\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}
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