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% do-vimlatex-onwrite
\documentclass[]{article}
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\newcommand{\herod}{
\text{herod}
}
\title{Domača naloga}
\author{Anton Luka Šijanec, 3. a}
\begin{document}
\maketitle
\begin{abstract}
Rešitev domače naloge, naročene 20. januarja 2022. Navodilo: \textbf{Mat3:} 43 / 3, 4, 9, 10. Ker na 43. strani sploh ni nalog 9 in 10, predvidevam, da je imel profesor v mislih naloge s strani 59.
\end{abstract}
% \tableofcontents
\section{Naloge in rešitve}
\begin{enumerate}[label=\textbf{\arabic*.}]
\setcounter{enumi}{2}
\item Marko je na dvorišču narisal trikotnik, katerega stranice so bile dolge \m{5}, \m{8} in \cm{12}, Tadeja pa je narisala trikotnik, katerega stranice so bile dolge \m{5}, \m{8} in \cm{9}. Kateri trikotnik je imel večjo ploščino? Za koliko?
$$
s = \frac{a+b+c}{2} \wedge s_M = \m{12,5} \wedge s_T = \m{11} \Rightarrow
S = \sqrt{s(s-a)(s-b)(s-c)} \wedge S_M = \kvm{210,9375} \wedge S_T = \kvm{396}
$$
Tadejin je večji za \kvm{185,0625}.
\item Stranici romboida sta dolgi \cm{52} in \cm{32}, diagonala pa \cm{44}. Kolikšna je njegova ploščina?
$$
s = \frac{\cm{52}+\cm{32}+\cm{44}}{2} \wedge
S=2\sqrt{s(s-\cm{52})(s-\cm{32})(s-\cm{44})}=\kvcm{1402,16974721}
$$
\setcounter{enumi}{8}
\item Stranice trikotnika so dolge \cm{5}, \cm{6} in \cm{7}. Koliko so veliki notranji koti tega trikotnika? Kolikšen je polmer trikotniku očrtane krožnice? Kolikšna je ploščina trikotnika?
$$
\gamma=\arccos\left(\frac{a^2+b^2-c^2}{2ab}\right)=\ang{78,463} \wedge
\alpha=\ang{57,122} \wedge \beta=\ang{44,415}
$$$$
R=\frac{c}{\sin\gamma}=2R
\wedge R=\frac{\frac{\cm{7}}{\sin\ang{78,463}}}{2}=\cm{3,572}
$$$$
S=\herod(\cm{5}, \cm{6}, \cm{7})=\kvcm{14,6969}
$$
\item Koliko sta dolgi najdaljša in najkrajša višina trikotnika, ki ima stranice dolge \cm{8}, \cm{11} in \cm{14}?
$$
v_a=\frac{2\herod(a, b, c)}{\cm{8}}=\cm{10,978} \wedge
v_b=\frac{2\herod(a, b, c)}{\cm{11}}=\cm{7,984} \wedge
v_c=\frac{2\herod(a, b, c)}{\cm{14}}=\cm{6,273}
$$
\end{enumerate}
\section{Zaključek}
\hologo{LaTeX} izvorna koda dokumenta je objavljena na \url{https://git.sijanec.eu/sijanec/sola-gimb-3}. Za izdelavo dokumenta je potreben \texttt{TeXLive 2020}.
\if\razhroscevanje1
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\section*{Razhroščevalne informacije}
Konec generiranja dokumenta \today\ ob \currenttime.
Dokument se je generiral R0qK1KR2 \SI{}{\second}. % aaasecgeninsaaa
\fi
\end{document}
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