Wiki-Quellcode von Lösung Dreieck, Seiten und Winkel
Verstecke letzte Bearbeiter
| author | version | line-number | content |
|---|---|---|---|
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2.1 | 1 | __Seitenlängen:__ |
| 2 | Seite {{formula}}\overline{AB}{{/formula}}: | ||
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1.1 | 3 | Zunächst stellen wir den Vektor {{formula}}\overrightarrow{AB}{{/formula}} auf: {{formula}}\overrightarrow{AB}=\left(\begin{matrix}7-(-1)\\1-(-1)\\\end{matrix}\right)=\left(\begin{matrix}8\\2\\\end{matrix}\right){{/formula}} |
| 4 | Nun berechnen wir die Seitenlänge: {{formula}}|\overrightarrow{AB}|=\sqrt{8^2+2^2}=\sqrt{68}\approx 8,25{{/formula}} | ||
| 5 | |||
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2.1 | 6 | Seite {{formula}}\overline{BC}{{/formula}}: |
| 7 | {{formula}}\overrightarrow{BC}=\left(\begin{matrix}1-7\\3-1\\\end{matrix}\right)=\left(\begin{matrix}-6\\2\\\end{matrix}\right){{/formula}} | ||
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1.1 | 8 | {{formula}}|\overrightarrow{BC}|=\sqrt{(-6)^2+2^2}=\sqrt{40}\approx 6,32{{/formula}} |
| 9 | |||
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2.1 | 10 | Seite {{formula}}\overline{AC}{{/formula}}: |
| 11 | {{formula}}\overrightarrow{AC}=\left(\begin{matrix}1-(-1)\\3-(-1)\\\end{matrix}\right)=\left(\begin{matrix}2\\4\\\end{matrix}\right){{/formula}} | ||
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1.1 | 12 | {{formula}}|\overrightarrow{BC}|=\sqrt{2^2+4^2}=\sqrt{20}\approx 4,47{{/formula}} |
| 13 | |||
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2.1 | 14 | __Innenwinkel:__ |
| 15 | Zur Berechnung der Innenwinkel verwenden wir die Formel | ||
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1.1 | 16 | |
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2.1 | 17 | {{formula}} |
| 18 | \begin{align} | ||
| 19 | &\cos(\alpha)=\frac{\vec{a}\cdot \vec{b}}{|\vec{a}|\cdot |\vec{b}|}\\ | ||
| 20 | \Leftrightarrow \ &\alpha=\cos^{-1}\left( \frac{\vec{a}\cdot \vec{b}}{|\vec{a}|\cdot |\vec{b}|} \right) | ||
| 21 | \end{align} | ||
| 22 | {{/formula}} | ||
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1.1 | 23 | |
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2.1 | 24 | Winkel zwischen {{formula}}\overrightarrow{AB}{{/formula}} und {{formula}}\overrightarrow{AC}{{/formula}}: |
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1.1 | 25 | |
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2.1 | 26 | {{formula}}\alpha_1=\cos^{-1}\left( \frac{\overrightarrow{AB}\cdot \overrightarrow{AC}}{|\overrightarrow{AB}|\cdot |\overrightarrow{AC}|} \right)=\cos^{-1}\left( \frac{\left(\begin{matrix}8\\2\\\end{matrix}\right)\cdot \left(\begin{matrix}2\\4\\\end{matrix}\right)}{\sqrt{68}\cdot \sqrt{20}} \right)=\cos^{-1}\left( \frac{8\cdot 2+2\cdot 4}{\sqrt{68}\cdot \sqrt{20}} \right)=\cos^{-1}\left( \frac{24}{\sqrt{1360}} \right)\approx 49,40^\circ{{/formula}} |
| 27 | |||
| 28 | |||
| 29 | Winkel zwischen {{formula}}\overrightarrow{BA}{{/formula}} und {{formula}}\overrightarrow{BC}{{/formula}}: | ||
| 30 | |||
| 31 | {{formula}}\alpha_2=\cos^{-1}\left( \frac{\overrightarrow{BA}\cdot \overrightarrow{BC}}{|\overrightarrow{BA}|\cdot |\overrightarrow{BC}|} \right)=\cos^{-1}\left( \frac{\left(\begin{matrix}-8\\-2\\\end{matrix}\right)\cdot \left(\begin{matrix}-6\\2\\\end{matrix}\right)}{\sqrt{68}\cdot \sqrt{40}} \right)=\cos^{-1}\left( \frac{(-8)\cdot(-6)+(-2)\cdot2}{\sqrt{68}\cdot \sqrt{40}} \right)=\cos^{-1}\left( \frac{44}{\sqrt{2720}} \right)\approx 32,47^\circ{{/formula}} | ||
| 32 |