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+% !TeX encoding = UTF-8
+% !TeX spellcheck = sl_SI
+% do-vimlatex-onwrite
+\documentclass[]{article}
+\usepackage[utf8]{inputenc}
+\usepackage{etoolbox}
+\usepackage[hidelinks]{hyperref}
+\usepackage[a4paper, total={7in, 10in}]{geometry}
+\usepackage{graphicx}
+\usepackage{hologo}
+\usepackage{amssymb}
+\usepackage{calculator}
+\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{dirtytalk}
+\usepackage{csquotes}
+\usepackage{listings}
+\usepackage[ddMMyyyy]{datetime}
+\usepackage{ccicons}
+\usepackage{chemformula}
+\usepackage{gensymb}
+\usepackage[normalem]{ulem}
+\usepackage[slovene]{babel}
+\usepackage{cancel}
+\usepackage{tabularx}
+\usepackage{xcolor}
+\usepackage{colortbl}
+\newcolumntype{Y}{>{\centering\arraybackslash}X}
+% \usepackage{multienum} % weird with labels
+\usetikzlibrary{calc} %% not really needed............. idk.
+%\usetikzlibrary{external}
+% \usetkzobj{all} % tkz-euclide > 3.02 tega ne potrebuje več (:
+%\tikzexternalize
+\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{codered}{rgb}{1,0,0}
+\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}
+\renewcommand{\dateseparator}{. }
+\settimeformat{hhmmsstime}
+\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{\snovdn}{Preverjanje znanja geometrije }
+\newcommand{\predmdn}{mat}
+\newcommand{\predmkaj}{domace\_naloge}
+\newcommand{\stevilkadn}{19}
+\newcommand{\cm}[1]{\SI{#1}{\centi\meter}}
+\newcommand{\kmh}[1]{\SI{#1}{\kilo\meter\per\hour}}
+\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{17. november 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}}
+\newcommand\hcancel[2][black]{\setbox0=\hbox{$#2$}%
+\rlap{\raisebox{.45\ht0}{\textcolor{#1}{\rule{\wd0}{1pt}}}}#2}
+% \everymath{\displaystyle} % https://tex.stackexchange.com/a/32847/212260
+\begin{document}
+\maketitle
+\begin{abstract}
+Ta dokument vsebuje domačo nalogo, ki zajema snov \textit{\snovdn}pri matematiki, in njene rešitve.
+\end{abstract}
+%\tableofcontents
+\section{Učni list \textit{\snovdn}}
+\begin{enumerate}[label=\textbf{\arabic*.}]
+ \item V enakokrakem trikotniku je zunanji kot ob osnovnici za \ang{24} manjši od zunanjega kota ob vrhu.
+ \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Izračunaj notranje kote tega trikotnika.
+ $$\ang{360}=2(\gamma'-\ang{24})+\gamma' \rightarrow \gamma'=\ang{104}
+ \rightarrow \gamma=\ang{76}\wedge\alpha=\ang{52} \wedge \beta=\ang{52}$$
+ \item Izračunaj kot med simetralama obeh zunanjih kotov ob osnovnici tega trikotnika.
+ $$\ang{180}-\ang{128}=\ang{52}$$
+ \item Izračunaj kot med višino na krak in višino na osnovnico tega trikotnika.
+ $$180-(\ang{180}-\ang{52}-\ang{90})=\ang{142}\rightarrow\ang{180}-\ang{142}=\underline{\underline{\ang{38}}}$$
+ \end{enumerate}
+ \item
+ \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Načrtaj trikotnik s podatki: $a=\cm{5}, t_c=\cm{4,5}, v_c=\cm{4}$.
+ \begin{center}
+ \begin{tikzpicture}
+ \tkzDefPoint(0,0){C} \tkzDefPoint(0,-4){V} \tkzDefPoint(0,-5){aH}
+ \tkzDefPoint(0,-4.5){tH}
+ \tkzCompass(C,V) \tkzLabelLine[below left](C,V){$v_c$}
+ \tkzDefMidPoint(C,V) \tkzGetPoint{M}\tkzCompass(C,M)\tkzCompass(V,M)
+ \tkzInterLC(C,V)(V,M) \tkzGetPoints{MAGAIN}{MX} \tkzCompass(V,MX)
+ \tkzDrawLine(C,MX) \tkzInterCC(M,MX)(MX,M) \tkzGetPoints{c}{d}
+ \tkzCompass(M,c)\tkzCompass(MX,c) \tkzCompass(M,d) \tkzCompass(MX,d)
+ \tkzInterLC(c,d)(C,aH) \tkzGetPoints{B}{BX} \tkzCompass(C,B)
+ \tkzDrawLine(C,B) \tkzLabelLine[below left](C,B){$a$}
+ \tkzInterLC(c,d)(C,tH) \tkzGetPoints{T}{TX} \tkzCompass(C,T)
+ \tkzDrawLine[red](C,T) \tkzLabelLine[below left, red](C,T){$t_c$}
+ \tkzDrawLine[blue](C,TX) \tkzLabelLine[left, blue](C,TX){$t_c$}
+ \tkzInterLC(c,d)(T,B) \tkzGetPoints{AX}{A} \tkzCompass(T,A)
+ \tkzDrawLine[red](A,C) \tkzLabelLine[below left, red](A,C){$b$}
+ \tkzInterLC(c,d)(TX,B) \tkzGetPoints{AX2}{A2} \tkzCompass(TX,A2)
+ \tkzDrawLine[blue](A2,C) \tkzLabelLine[below, blue](A2,C){$b'$}
+ \tkzLabelLine[below left, red](A,B){$c$} \tkzDrawLine(c,A2)
+ \tkzLabelLine[below left, blue](A2,B){$c$}
+ \tkzCompass[blue](C,TX) \tkzLabelPoint[blue](A2){$A$}
+ \tkzDrawPolygon[fill=blue!30, opacity=0.30](A2,B,C)
+ \tkzDrawPolygon[fill=red!30, opacity=0.30](A,B,C)
+ \tkzDrawPoints(C,B) \tkzLabelPoints[below left](C,B)
+ \tkzDrawPoints[blue](A2) \tkzDrawPoints[red](A)
+ \tkzLabelPoints[red, below left](A)
+ \end{tikzpicture}
+ \end{center}
+ \begin{center}
+ Obstajata \color{red}dve \color{blue}rešitvi\color{black}.
+ \end{center}
+ \item Načrtaj romb s podatki: $e+f=\cm{16}, \alpha=\ang{60}$.
+ \begin{center}
+ \begin{figure}[H]
+ \hspace*{-2cm}
+ \begin{tikzpicture}
+ \tkzDefPoint(0,0){O} \tkzDefPoint(16,0){C}
+ \tkzDrawLine(O,C) \tkzLabelLine[below](O,C){$\cm{16}$}
+ \tkzInterCC(O,C)(C,O) \tkzGetPoints{6}{7}
+ \tkzCompass(O,6) \tkzCompass(O,7) \tkzCompass(C,6) \tkzCompass(C,7)
+ \tkzDefMidPoint(6,C) \tkzGetPoint{3}\tkzCompass(6,3)\tkzCompass(C,3)
+ \tkzDrawLine(O,3) \tkzInterLC(O,C)(O,3) \tkzGetPoints{USELESS}{4}
+ \tkzCompass(O,4) \tkzInterCC(3,4)(4,3) \tkzGetPoints{USELESS}{1}
+ \tkzCompass(3,1)\tkzCompass(4,1)\tkzDefMidPoint(O,6) \tkzGetPoint{2}
+ \tkzCompass(O,2)\tkzCompass(6,2)\tkzDrawLine(C,2)
+ \tkzInterLL(O,1)(C,2) \tkzGetPoint{D} \tkzDrawLine(O,D)
+ \tkzInterLC(O,C)(D,C) \tkzGetPoints{USELESS}{A} \tkzDrawLine(A,D)
+ \tkzCompass(D,A) \tkzInterCC(A,D)(C,D) \tkzGetPoints{USELESS}{B}
+ \tkzCompass(A,B) \tkzCompass(C,B) \tkzDrawLine(A,B)\tkzDrawLine(C,B)
+ \tkzDrawPolygon[fill=red!30, opacity=.3](A,B,C,D)
+ \tkzLabelLine[below left](A,B){$a$}
+ \tkzLabelLine[below](C,B){$a$}
+ \tkzLabelLine[below left](C,D){$a$}
+ \tkzLabelLine[left](A,D){$a$}
+ \tkzLabelPoints[below left](O,C,D,A,B) \tkzDrawPoints(O,C,D,A,B)
+ \end{tikzpicture}
+ \end{figure}
+ \end{center}
+ \end{enumerate}
+ \item
+ \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Dokaži, da je paralelogram, v katerem sta diagonali enako dolgi, pravokotnik.
+ $$
+ AC=BD\wedge BC=AD\rightarrow\bigtriangleup ABC\Tilde=\bigtriangleup BAD\rightarrow\measuredangle ABC=\measuredangle BAD\wedge\measuredangle ABC+\measuredangle BAD=\ang{180}\rightarrow
+ $$$$
+ 2\cdot\measuredangle ABC=\ang{180}\rightarrow\measuredangle ABC=\ang{180}
+ $$
+ \item Enakokraki trapez z osnovnicama $a=\cm{18}$ in $c=\cm{8}$ je tangenten. Izračunaj dolžino višine na osnovnico.
+ $$
+ b=d\wedge\alpha=\beta\rightarrow\frac{a+c}{2}=b=\cm{13}\rightarrow
+ v_c=\sqrt{\cm{13}^2-\left(\frac{a-c}{2}\right)^2}=
+ $$$$
+ \sqrt{\cm{13}^2-\cm{5}^2}=\sqrt{\cm{144}^2}=\cm{12}
+ $$
+ \end{enumerate}
+ \item
+ \begin{enumerate}[label=\textbf{\xslalph*)}]
+ \item Naj bodo $A_1, A_2, ... A_9$ oglišča pravilnega devetkotnika.
+ \begin{enumerate}[label=\textbf{a_{\arabic*})}]
+ \item Izračunaj kot med daljicama $A_{1}A_{2}$ in $A_{2}A_{3}$.
+ $$\Sigma_\text{notranjih kotov}=
+ \left(n_{=8}\right)-2)\cdot\ang{180}=\ang{1080}
+ \rightarrow\alpha=\frac{\ang{1080}}{8}=\ang{135}$$
+ \item Izračunaj kot med daljicama $A_{1}A_{3}$ in $A_{3}A_{5}$.
+ $$\Sigma_\text{notranjih kotov}=
+ \left(n_{=4}\right)-2)\cdot\ang{180}=\ang{360}
+ \rightarrow\alpha=\frac{\ang{360}}{4}=\ang{90}$$
+ \end{enumerate}
+ \item Če enakokrakemu trikotniku $ABC$ očrtamo krožnico, pripada kraku $AC$ središčni kot \ang{75}. Izračunaj kote tega trikotnika.
+ $$\beta=\alpha=\frac{\ang{75}}{2}=\ang{37,5}\rightarrow\gamma=\ang{180}-2\cdot\ang{37,5}=\ang{105}$$
+ \end{enumerate}
+ \item Dokaži, da se v deltoidu diagonali sekata pod pravim kotom.
+ $$
+ \bigtriangleup ABD\Tilde=\bigtriangleup BCD\rightarrow
+ \overline{AP_{\left(\text{presečišče diagonal}\right)}}_{\left(v_A\right)}
+ =\overline{PC}_{\left(v_C\right)}\rightarrow
+ $$
+ $$
+ \bigtriangleup APD\Tilde=\bigtriangleup PCD\wedge P\in AC\wedge
+ \measuredangle CPD=\measuredangle DPA=\measuredangle APB=\measuredangle BPC
+ \rightarrow
+ $$
+ $$
+ \measuredangle CPD=\measuredangle DPA=\measuredangle APB=\measuredangle BPC
+ = \ang{90}
+ $$
+\end{enumerate}
+\begin{tabularx}{\textwidth}{|c||*{6}{Y|}}
+ \hline
+ \cellcolor{yellow}nadloga & \cellcolor{yellow}1 & \cellcolor{yellow}2 & \cellcolor{yellow}3 & \cellcolor{yellow}4 & \cellcolor{yellow}5 \\
+ \hline
+ \textbf{točke} & $2+2+2=\mathbf{6}$ & $3+3=\mathbf{6}$ & $3+3=\mathbf{6}$ & $(1,5+1,5)+3=\mathbf{6}$ & $\mathbf{8}$ \\
+ \hline
+\end{tabularx}
+\begin{center}
+ Rešiti je treba \textbf{ŠTIRI} nadloge.
+
+ $50 \% = 13$ točk, $100 \% = 26$ točk
+\end{center}
+\section{Zaključek}
+Ta dokument je informativne narave in se lahko še spreminja. Najnovejša različica, torej PDFji in
+\hologo{LaTeX}\footnote{Za izdelavo dokumenta potrebujete \texttt{f5ff} skript Antona Luke Šijanca, \texttt{TeXLive 2020} in operacijski sistem \texttt{DebIan/GNU/Linux}. Za urejanje priporočam urejevalnik besedil \texttt{vim} in pregledovalnik PDF dokumentov \texttt{evince}.}
+izvorna koda, zgodovina sprememb in prejšnje različice, je na voljo v mojem šolskem Git repozitoriju na
+\url{https://git.sijanec.eu/sijanec/sola-gimb-2} v mapi
+\href{https://git.sijanec/sola-gimb-2/src/branch/master/\predmdn/\predmkaj/\stevilkadn/}{/\predmdn/\predmkaj/\stevilkadn/}. Povezava za ogled zadnje različice tega dokumenta v PDF obliki je \url{http://razor.arnes.si/~asija3/files/sola/gimb/2/\predmdn/\predmkaj/\stevilkadn/dokument.pdf} in/ali \url{https://git.sijanec.eu/sijanec/sola-gimb-2/raw/branch/master/\predmdn/\predmkaj/\stevilkadn/dokument.pdf}.
+
+\if\razhroscevanje1
+\section*{Razhroščevalne informacije}
+Te informacije so generirane, ker je omogočeno razhroščevanje. Pred 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: \today\ ob \currenttime\footnote{To ne nakazuje dejanskega časa, ko je bil dokument napisan, temveč čas, ko je bi dokument generiran v PDF/DVI obliko. Isto velja za datum v glavi dokumenta. Če berete direktno iz LaTeX datoteke, bo to vedno današnji datum.}%\input|"date -Ins"
+
+Dokument se je generiral R0qK1KR2 \SI{}{\second}.
+\fi
+% \item $$$$ aaasecgeninsaaa R0qK1KR2
+\end{document}
generated by cgit v1.2.3 (git 2.25.1) at 2024-11-21 17:11:49 +0100