path: root/mat/domace_naloge/8/dokument.tex

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% \usetkzobj{all} % tkz-euclide > 3.02 tega ne potrebuje več (:
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			\begin{tabular}[t]{c}% <------
				\@author%            <------ Authors
			\end{tabular}\par}%    <------
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		{\large \@date}%
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	\par
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%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}
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	\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}%
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\renewcommand\abstractname{Povzetek}
\date{30. september 2020}
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\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}