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-rw-r--r--mat/domace_naloge/5/dokument.aux20
-rw-r--r--mat/domace_naloge/5/dokument.log1399
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-rw-r--r--mat/domace_naloge/5/dokument.tex286
-rw-r--r--mat/domace_naloge/5/dokument.toc2
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+\BOOKMARK [1][-]{section.1}{Ucni list Merjenje, trikotni\235ka neenakost : Naloge 1 do 12}{}% 1
+\BOOKMARK [1][-]{section.2}{Zakljucek}{}% 2
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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
+\newcommand{\stevilkadn}{5}
+\newcommand{\snovdn}{Merjenje, trikotniška neenakost }
+\makeatletter
+\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{16. 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 15. septembra 2020, in njihove
+ rešitve, ki sem jih spisal sam. Naloge obsegajo snov \textit{\snovdn}in so
+ iz učnega lista. Kjer je bilo potrebno izbrati neke poljubne naloge,
+ sem jih vedno izbral naključno.
+\end{abstract}
+\tableofcontents
+
+\section{Učni list \textit{\snovdn}: Naloge 1 do 12}
+\begin{enumerate}[label=\textbf{\arabic*.}] % 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
+ \item \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Na premici $p$ leži točka $T$. Opiši množico vseh tistih točk na premici, ki so od $T$ oddaljene za 3 dolžinske enote.
+ $$m=(X; d(T,X) = 3 \wedge X \in p)$$
+ \item Izmeri daljici $AB$ in $CD$ in načrtaj daljico $EF$, za katero velja:
+ \begin{enumerate}[label=\textbf{b\arabic*)}]
+ \begin{multicols}{2}
+ \item $|EF|=|AB|+|CD|$
+ \begin{center}
+ \begin{tikzpicture}
+ \tkzDefPoint(0,0){A}
+ \tkzDefPoint(1,0){B}
+ \tkzDefPoint(1,0){C}
+ \tkzDefPoint(2,0){D}
+ \tkzDefPoint(0,0){E}
+ \tkzDefPoint(2,0){F}
+ \tkzDrawPoints(A,B,C,D,E,F)
+ \tkzLabelPoints[above right](A,B,D)
+ \tkzLabelPoints[below left](E,F,C)
+ \tkzDrawLine(C,D)
+ \tkzDrawLine(A,B)
+ \end{tikzpicture}
+ \end{center}
+ \item $|EF|=2|AB|+|CD|$
+ \begin{center}
+ \begin{tikzpicture}
+ \tkzDefPoint(0,0){A}
+ \tkzDefPoint(1,0){B}
+ \tkzDefPoint(1,0){C}
+ \tkzDefPoint(2,0){D}
+ \tkzDefPoint(-1,0){E}
+ \tkzDefPoint(2,0){F}
+ \tkzDrawPoints(A,B,C,D,E,F)
+ \tkzLabelPoints[above right](A,B,D)
+ \tkzLabelPoints[below left](E,F,C)
+ \tkzDrawLine(E,F)
+ \tkzCompass(B,E)
+ \end{tikzpicture}
+ \end{center}
+ \end{multicols}
+ \end{enumerate}
+
+ \end{enumerate}
+
+ \item Naslednje kote zapiši s stopinjami, minutami in sekundami.
+ \begin{multicols}{2}
+ \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item $\alpha=\ang{23.58}=\ang{23}0,58\cdot60'=\ang{23}34'0,8\cdot60''=
+ \mathbf{\ang{23}34'48''}$
+ \item $\beta=\ang{78.409}=\ang{78}0,409\cdot60'=
+ \ang{78}24'0,54\cdot60''=\mathbf{\ang{78}24'32,4''}$
+ \end{enumerate}
+ \end{multicols}
+ \item Naslednjim kotom izračunaj komplementarne in suplementarne kote:
+ \begin{multicols}{2}
+ \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item $\alpha=\ang{23}$
+ \begin{center}
+ $$\text{komplementarni: } \ang{90}-\ang{23}=\ang{67}$$
+ $$\text{suplementarni/sokot: } \ang{180}-\ang{23}=\ang{157}$$
+ \end{center}
+ \item $\beta=\ang{78.49}$
+ \begin{center}
+ $$\text{komplementarni: } \ang{90}-\ang{78.49}=\ang{11.51}$$
+ $$\text{suplementarni/sokot: } \ang{180}-\ang{78.49}=\ang{101.51}$$
+ \end{center}
+ \end{enumerate}
+ \end{multicols}
+ \item \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Razlika dveh sokotov je \ang{23,58}. Koliko merita oba kota? Rezultata naj bosta vsaj do kotne minute natančna.
+ $$\ang{23.58}=(\ang{180}-\alpha)-\alpha \rightarrow
+ 2\alpha=\ang{180}-\ang{23,58}\rightarrow\alpha=\frac{\ang{180}-\ang{23,58}}{2}=\ang{78.21}\rightarrow\beta=\ang{180}-\alpha=\ang{101.79}$$
+ \item Razlika dveh komplementarnih kotov je $\ang{23}58'$. Koliko merita oba kota? Rezultata naj bosta vsaj do kotne minute natančna.
+ $$\alpha=\frac{\ang{180}-\ang[parse-numbers=false]{23,9\overline{6}}}{2}=\ang[parse-numbers=false]{78,01\overline{6}} \rightarrow \beta=\ang{180}-\alpha=\ang{180}-\ang[parse-numbers=false]{78,01\overline{6}}=\ang[parse-numbers=false]{101,98\overline{3}}$$
+ \item Vsota dveh kotov je $\ang{146}15'$, njuna razlika pa $\ang{23}38'$. Koliko merita oba kota? Rezultata naj bosta do kotne minute natančna.
+ $$(\alpha-\ang[parse-numbers=false]{23,6\overline{3}})+\alpha=\ang{146,25}\rightarrow2\alpha=\ang{146,25}-\ang[parse-numbers=false]{23,6\overline{3}}\rightarrow\alpha=\frac{\ang{146,25}-\ang[parse-numbers=false]{23,6\overline{3}}}{2}=\ang[parse-numbers=false]{61,308\overline{3}}\rightarrow$$
+ $$\beta=\ang{146,25}-\alpha=\ang[parse-numbers=false]{84,941\overline{6}}$$
+ \end{enumerate}
+ \item \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Koliko meri kot, če je petkratnik svojega komplementarnega kota?
+ $$\frac{5}{6}\ang{90}=\ang{75}$$
+ \item Koliko meri kot, če je šestkratnik svojega suplementarnega kota? Rezultat naj bo vsaj do kotne minute natančen.
+ $$\frac{6}{7}\ang{180}=\ang[parse-numbers=false]{154.\overline{285714}}$$
+ \item Od dveh sokotov je eden za $\frac{3}{7}$ pravega kota večji od drugega. Koliko meri eden in drugi? Rezultat naj bo vsaj do kotne minute natančen.
+ $$(\alpha+\frac{3}{7}\ang{90})+\alpha=\ang{180}\rightarrow
+ \alpha=\frac{\ang{180}-\frac{3}{7}\ang{90}}{2}=\ang[parse-numbers=false]{19,\overline{285714}}\rightarrow\beta=\ang{180}-\alpha=\ang{180}-\ang[parse-numbers=false]{19,\overline{285714}}=\ang[parse-numbers=false]{160,\overline{7142857}}$$
+ \item Eden od dveh komplementarnih kotov je 4-kratnik drugega. Koliko meri eden in drugi?
+ \begin{multicols}{2}
+ $$\alpha=\frac{4}{5}\ang{90}=\ang{72}$$
+
+ $$\beta=\frac{1}{5}\ang{90}=\ang{18}$$
+ \end{multicols}
+ \item Kolikšna je vsota sokotov dveh komplementarnih kotov?
+ $$\ang{90}$$
+ \item Poišči tak par suplementarnih kotov, katerih razlika je pravi kot.
+ $$(\alpha+\ang{90})+\alpha=\ang{180}\rightarrow\alpha=\frac{\ang{180}-\ang{90}}{2}=\ang{45}\rightarrow\beta=\ang{180}-\alpha=\ang{180}-\ang{45}=\ang{135}$$
+ \item Kota $\alpha=3x$ in $\beta=x+\ang{20}$ sta komplementarna. Poišči ju.
+ $$3x+x+\ang{20}=\ang{90}=4x+\ang{20}\rightarrow x=\frac{\ang{90}-\ang{20}}{4}=\ang{17.5}\rightarrow\alpha=3x=\ang{52.5}\wedge\beta=x+\ang{20}=\ang{37.5}$$
+ \end{enumerate}
+ \iffalse % nedokončano ˇˇˇ
+ \item \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Dve premici se sekata. Dokaži, da oba para nesosednjih kotov merita enako.
+ \begin{center}
+ trenutno mi ne kane na misel ...
+ \end{center}
+ \item Tri premice se sekajo v skupni točki. Dokaži, da je vsota treh nesosednjih kotov enaka \ang{180}.
+ \begin{center}
+ tudi tole ne ...
+ \end{center}
+ \item Skozi vrh pravega kota potegnem premico tako, da dobimo štiri kote. Pokaži, da je razlika dveh nesosednjih kotov enaka \ang{90}.
+ \begin{center}
+ jutri dalje, prepozno je, da bi moji možgani uporabno funkcionirali ...
+ \end{center}
+ \end{enumerate}
+ \item Konstruiraj, kjer je to mogoče, trikotnik s podatki:
+ \begin{multicols}{2}
+ \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item $a=\SI{7}{\centi\meter}, b=\SI{4}{\centi\meter}, c=\SI{5}{\centi\meter}$
+ \item $a=\SI{7}{\centi\meter}, b=\SI{4}{\centi\meter}, c=\SI{8}{\centi\meter}$
+ \item $a=\SI{7}{\centi\meter}, b=\SI{4}{\centi\meter}, c=\SI{13}{\centi\meter}$
+ \item $a=\SI{7}{\centi\meter}, b=\SI{4}{\centi\meter}, c=\SI{2}{\centi\meter}$
+ \item $a=\SI{15}{\centi\meter}, b=\SI{8}{\centi\meter}, c=\SI{23}{\centi\meter}$
+ \item $a=\SI{14}{\centi\meter}, b=\SI{8}{\centi\meter}, c=\SI{23}{\centi\meter}$
+ \end{enumerate}
+ \end{multicols}
+ \item \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Točki $A$ in $B$ sta oddaljeni 5. Ali obstaja točka, ki je od $A$ oddaljena 12, od $B$ pa 6?
+ \item Točki $A$ in $B$ na ravnini sta oddaljeni za 5. Opiši, kje leži točka $C$, ki je od točke $A$ oddaljena 3, hkrati pa najbližja točki $B$.
+ \end{enumerate}
+ \item Ali obstajajo trikotniki z naslednjimi dolžinami stranic?
+ \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item $a=\SI{8}{\centi\meter}, b=\SI{5}{\centi\meter}, c=\SI{14}{\centi\meter}$
+ \item $a=\SI{8}{\centi\meter}, b=\SI{15}{\centi\meter}, c=\SI{32}{\centi\meter}$
+ \end{enumerate}
+ \item \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Palico prelomiš na tri dele tako, da je vsak naslednji del polovico krajši od prejšnjega. Ali lahko iz teh treh delov sestaviš trikotnik?
+ \item[\textbf{* b)}] Palico prelomiš na tri dele tako, da je vsak naslednji del za desetino dolžine celotne palice krajši od prejšnjega. Ali lahko iz teh treh delov sestaviš trikotnik?
+ \end{enumerate}
+ \item[\textbf{* 11.}] Naj bo $D$ poljubno izbrana točka na stranici $BC$ trikotnika $\bigtriangleup ABC$. Dokaži:
+ $$d(A,D)+d(D,B)\leq d(A,C) + d(C,B)$$
+ \setcounter{enumi}{11}
+ \item Če se komu zdi, da je v matematiki odveč zahtevati komutativnost razdalje med točkama $(d(A,B)=d(B,A))$, ker da je to pač \textit{"samo po sebi umevno"}, naj pomisli: Dostikrat merimo ratdaljo dveh krajev A in B s časom, ki ga potrebujemo, da pridemo iz enega kraja v drugi kraj. In če je, denimo, kraj A v dolini, kraj B pa vrh gore, je zagotovo:
+ $$d(A,B)\not=d(B,A)$$
+ \fi % ^^^ nedokončano
+\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}
diff --git a/mat/domace_naloge/5/dokument.toc b/mat/domace_naloge/5/dokument.toc
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+\contentsline {section}{\numberline {1}Učni list \textit {Merjenje, trikotniška neenakost }: Naloge 1 do 12}{1}{section.1}%
+\contentsline {section}{\numberline {2}Zaključek}{2}{section.2}%
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