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diff --git a/mat/domace_naloge/7/dokument.tex b/mat/domace_naloge/7/dokument.tex new file mode 100644 index 0000000..d3a3c3b --- /dev/null +++ b/mat/domace_naloge/7/dokument.tex @@ -0,0 +1,327 @@ +% !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} |