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Eksāmens VĒSTURĒ 12. KLASEI
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### Teorija

 Trijstūris Trijstūris Trijstūris $\begin{array}{l}{S}_{\mathrm{\Delta }}=\frac{a\cdot {h}_{a}}{2}\\ {S}_{\mathrm{\Delta }}=\frac{1}{2}\mathit{absin}\mathrm{\gamma }\\ {S}_{\mathrm{\Delta }}=\frac{\mathit{abc}}{4R}\\ \\ {S}_{\mathrm{\Delta }}=p\cdot r\\ \\ {S}_{\mathrm{\Delta }}=\sqrt{p\left(p-a\right)\left(p-b\right)\left(p-c\right)}\end{array}$ Trijstūris$\begin{array}{l}\frac{a}{\mathit{sin}\mathrm{\alpha }}=\frac{b}{\mathit{sin}\mathrm{\beta }}=\frac{c}{\mathit{sin}\mathrm{\gamma }}=2R\\ R=\frac{a}{2\mathit{sin}\mathrm{\alpha }}\\ {a}^{2}={b}^{2}+{c}^{2}-2\mathit{bc}\cdot \mathit{cos}\mathrm{\alpha }\end{array}$ Bisektrises īpašība $\frac{\mathit{AD}}{\mathit{DC}}=\frac{\mathit{AB}}{\mathit{BC}}$ Mediānu īpašība$\frac{\mathit{BO}}{\mathit{OD}}=\frac{\mathit{AO}}{\mathit{OF}}=\frac{\mathit{CO}}{\mathit{OE}}=\frac{2}{1}$ Viduslīnijas īpašība $\mathit{ED}=\frac{1}{2}\mathit{BC}$ Taisnleņķa trijstūris Regulārs trijstūris Līdzīgi trijstūri ${h}_{c}$ - augstums pret hipotenūzu${a}_{c}$, ${b}_{c}$ - katešu projekcijas uz hipotenūzas$\begin{array}{l}{h}_{c}^{2}={a}_{c}\cdot {b}_{c}\\ {a}^{2}={a}_{c}\cdot c\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{b}^{2}={b}_{c}\cdot c\\ \frac{{a}^{2}}{{b}^{2}}=\frac{{a}_{c}}{{b}_{c}}\end{array}$ $\begin{array}{l}h=\frac{a\sqrt{3}}{2}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}r=\frac{a\sqrt{3}}{6}\\ R=\frac{a\sqrt{3}}{3}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}S=\frac{{a}^{2}\sqrt{3}}{4}\end{array}$ $\begin{array}{l}\frac{\mathit{AB}}{{A}_{1}{B}_{1}}=\frac{\mathit{AC}}{{A}_{1}{C}_{1}}=\frac{\mathit{BC}}{{B}_{1}{C}_{1}}=\frac{{P}_{}}{{P}_{1}}=k\\ \\ \frac{{S}_{\mathit{ABC}}}{{S}_{{A}_{1}{B}_{1}{C}_{1}}}={k}^{2}\end{array}$ Paralelograms Ievilkti un apvilkti četrstūri Trapece $\begin{array}{l}2\left({a}^{2}+{b}^{2}\right)={d}_{1}^{2}+{d}_{2}^{2}\\ S=a\cdot h\phantom{\rule{0.147em}{0ex}}=\mathit{absin}\mathrm{\alpha }\end{array}$ Ievilkts četrstūris $\mathit{ABCD}$$\sphericalangle A+\sphericalangle C=\sphericalangle B+\sphericalangle D$ Apvilkts četrstūris $\mathit{ABCD}$$\mathit{AB}+\mathit{CD}=\mathit{AD}+\mathit{BC}$ $S=\frac{a+b}{2}\cdot h$

 Nogriežņi un leņķi, kas saistīti ar riņķa līniju Regulāri $$n$$-stūri $\begin{array}{l}\sphericalangle \mathit{BSA}=\frac{1}{2}\left(\cup \mathit{BA}+\cup \mathit{CD}\right)\phantom{\rule{1.323em}{0ex}}\sphericalangle \mathit{EAC}=\frac{1}{2}\left(\cup \mathit{FD}-\cup \mathit{EC}\right)\\ \mathit{AS}\cdot \mathit{SC}=\mathit{BS}\cdot \mathit{SD}\phantom{\rule{3.234em}{0ex}}\sphericalangle \mathit{FBG}=\frac{1}{2}\cup \mathit{FB}\\ \phantom{\rule{8.379em}{0ex}}{\mathit{AB}}^{2}=\mathit{AC}\cdot \mathit{AD}\\ \phantom{\rule{8.379em}{0ex}}\mathit{AE}\cdot \mathit{AF}=\mathit{AC}\cdot \mathit{AD}\end{array}$ $\begin{array}{l}S=\frac{1}{2}P\cdot r\\ \\ {a}_{n}=2R\cdot \mathit{sin}\frac{{180}^{o}}{n}\\ \\ {a}_{n}=2r\cdot \mathit{tg}\frac{{180}^{o}}{n}\end{array}$

 Prizma Konuss Riņķis un riņķa līnija $V=S\cdot H$ Cilindrs$\begin{array}{l}{S}_{s}=2\mathrm{\pi }\cdot R\cdot H\\ V=\mathrm{\pi }\cdot {R}^{2}\cdot H\end{array}$ $\begin{array}{l}{S}_{s}=\mathrm{\pi }\cdot R\cdot l\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{S}_{s}=\frac{\mathrm{\pi }{l}^{2}\mathrm{\alpha }}{{360}^{o}}\\ V=\frac{\mathrm{\pi }{R}^{2}H}{3}\\ \mathit{No}šķ\mathit{elts}\phantom{\rule{0.147em}{0ex}}\mathit{konuss}\end{array}$ Nošķelts konuss$\begin{array}{l}{S}_{s}=\mathrm{\pi }\left({R}_{1}+{R}_{2}\right)\cdot l\\ V=\frac{\mathrm{\pi }H}{3}\left({R}_{1}^{2}+{R}_{1}{R}_{2}+{R}_{2}^{2}\right)\end{array}$ $\begin{array}{l}C=2\mathrm{\pi }R\phantom{\rule{0.735em}{0ex}}{I}_{\mathrm{\alpha }}=\frac{\mathrm{\pi }\cdot r\cdot \mathrm{\alpha }}{{180}^{o}}\phantom{\rule{0.294em}{0ex}}\\ \\ S=\mathrm{\pi }{R}^{2}\phantom{\rule{0.735em}{0ex}}{S}_{\mathit{sekt}}=\frac{\mathrm{\pi }\cdot {r}^{2}\cdot \mathrm{\alpha }}{{360}^{o}}\end{array}$ Piramīda Lode un tās daļas Vektori $\begin{array}{l}{S}_{s.\mathit{reg}.}=\frac{1}{2}P\cdot {h}_{s}\\ {S}_{s.\mathit{reg}.}=\frac{{S}_{\mathit{pamata}}}{\mathit{cos}\mathrm{\alpha }}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \phantom{\rule{0.147em}{0ex}}V=\frac{1}{3}S\cdot H\end{array}$ Nošķelta piramīda$\begin{array}{l}{S}_{s.\mathit{reg}.n.}=\frac{1}{2}\left({P}_{1}+{P}_{2}\right)\cdot {h}_{s}\\ V=\frac{H}{3}\left({S}_{1}+{S}_{2}+\sqrt{{S}_{1}{S}_{2}}\right)\end{array}$ $\begin{array}{l}S=4\mathrm{\pi }{R}^{2}\phantom{\rule{0.441em}{0ex}}V=\frac{4}{3}\mathrm{\pi }{R}^{3}\\ {S}_{\mathit{segm}.\mathit{sf}.\mathit{virsma}}=2\mathrm{\pi }\mathit{RH}\\ {V}_{\mathit{segmentam}}=\mathrm{\pi }{H}^{2}\left(R-\frac{H}{3}\right)\\ {V}_{\mathit{sekt}}=\frac{2}{3}\mathrm{\pi }{R}^{2}H\phantom{\rule{0.147em}{0ex}}\end{array}$$H$ - segmenta augstums $\begin{array}{l}A\left({x}_{1};{y}_{1}\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}B\left({x}_{2};{y}_{2}\right)\\ \\ \stackrel{\to }{\mathit{AB}}=\left({x}_{2}-{x}_{1};{y}_{2}-{y}_{1}\right)\\ \stackrel{\to }{a}=\left({a}_{x};{a}_{y}\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\stackrel{\to }{b}=\left({b}_{x};{b}_{y}\right)\\ \\ \stackrel{\to }{a}+\stackrel{\to }{b}=\left({a}_{x}+{b}_{x};{a}_{y}+{b}_{y}\right)\\ \stackrel{\to }{a}-\stackrel{\to }{b}=\left({a}_{x}-{b}_{x};{a}_{y}-{b}_{y}\right)\\ \\ \left|\stackrel{\to }{a}\right|=\sqrt{{a}_{x}^{2}+{a}_{y}^{2}}\text{}\end{array}$