$\underset{‾}{\text{Berechne:}}\backslash underline\{\backslash text\{\backslash large\{Berechne:\}\}\}$

$1.)\text{L}\ddot{\text{a}}\text{nge der Strecke}[AE]\text{ in dm}2.)\text{L}\ddot{\text{a}}\text{nge der Strecke}[EF]\text{ in dm}3.)\text{L}\ddot{\text{a}}\text{nge der Strecke}[BD]\text{ in dm}4.)\text{Fl}\ddot{\text{a}}\text{cheninhalt des Trapezes ABDE}{\text{in dm}}^{2}1.)\backslash \; \backslash text\{Länge\backslash \; der\backslash \; Strecke\}\backslash \; [AE]\backslash \; \backslash text\{in\backslash \; dm\}\backslash \backslash 2.)\backslash \; \backslash text\{Länge\backslash \; der\backslash \; Strecke\}\backslash \; [EF]\backslash \; \backslash text\{in\backslash \; dm\}\backslash \backslash 3.)\backslash \; \backslash text\{Länge\backslash \; der\backslash \; Strecke\}\backslash \; [BD]\backslash \; \backslash text\{in\backslash \; dm\}\backslash \backslash 4.)\backslash \; \backslash text\{Flächeninhalt\backslash \; des\backslash \; Trapezes\backslash \; ABDE\}\backslash \; \backslash text\{in\backslash \; dm\}^2$

$\backslash rule\{10cm\}\{0.5mm\}$

$\text{zu 1.) }\sin {60}^{\circ }={\displaystyle \frac{\overline{AE}}{16dm}}\Rightarrow \overline{AE}=\sin {60}^{\circ }\cdot 16dm\backslash text\{zu\backslash \; 1.)\}\backslash \; \backslash sin60^\{\backslash circ\}\backslash \; =\backslash \; \backslash dfrac\{\backslash overline\{\{AE\}\}\}\{16\backslash \; dm\}\backslash quad\backslash Rightarrow\backslash quad\backslash \; \backslash overline\{AE\}\backslash \; =\backslash \; \backslash sin60^\{\backslash circ\}\backslash cdot16\backslash \; dm$

$\underset{‾}{\overline{AE}\approx \mathrm{13,9}dm}\backslash hspace\{48mm\}\backslash underline\{\backslash overline\{AE\}\backslash \; \backslash approx\backslash \; 13\{,\}9\backslash \; dm\}$

$\backslash rule\{10cm\}\{0.5mm\}$

$\text{zu 2.) }\cos {60}^{\circ }={\displaystyle \frac{\overline{EF}}{16dm}}\Rightarrow \overline{EF}=\cos {60}^{\circ }\cdot 16dm\backslash text\{zu\backslash \; 2.)\}\backslash \; \backslash cos60^\{\backslash circ\}\backslash \; =\backslash \; \backslash dfrac\{\backslash overline\{\{EF\}\}\}\{16\backslash \; dm\}\backslash quad\backslash Rightarrow\backslash quad\backslash \; \backslash overline\{EF\}\backslash \; =\backslash \; \backslash cos60^\{\backslash circ\}\backslash cdot16\backslash \; dm$

$\underset{‾}{\overline{EF}=8dm}\backslash hspace\{48mm\}\backslash underline\{\backslash overline\{EF\}\backslash \; =\backslash \; 8\backslash \; dm\}$

$\backslash rule\{10cm\}\{0.5mm\}$

$\text{zu 3.) }{\overline{BD}}^2=(\overline{EF}+\overline{DE})\cdot \overline{DC}\text{H}\ddot{\text{o}}\text{hensatz}\backslash text\{zu\backslash \; 3.)\}\backslash \; \backslash overline\{BD\}^2\backslash \; =\backslash \; (\backslash overline\{EF\}+\backslash overline\{DE\})\backslash cdot\backslash overline\{DC\}\backslash qquad\; \backslash large\{\backslash text\{\backslash color\{green\}Höhensatz\}\}$

$\overline{BD}=\sqrt{(8dm+22dm)\cdot 2dm}\backslash hspace\{9mm\}\backslash overline\{BD\}\backslash \; =\backslash \; \backslash sqrt\{(8\backslash \; dm+22\backslash \; dm)\backslash cdot\backslash \; 2\backslash \; dm\}$

$\underset{‾}{\overline{BD}\approx \mathrm{7,7}dm}\backslash hspace\{9mm\}\backslash underline\{\backslash overline\{BD\}\backslash \; \backslash approx\backslash \; 7\{,\}7\backslash \; dm\}$

$\backslash rule\{10cm\}\{0.5mm\}$

$\text{zu 4.) }{A}_{\text{ABDE}}={\displaystyle \frac{(\overline{AE}+\overline{BD})}{2}}\cdot \overline{ED}\Rightarrow A_{\text{ABDE}}={\displaystyle \frac{(\mathrm{13,9}dm+\mathrm{7,7}dm)}{2}}\cdot 22dm\backslash text\{zu\backslash \; 4.)\}\backslash \; A\_\backslash text\{\{ABDE\}\}\backslash \; =\backslash \; \backslash dfrac\{(\backslash overline\{AE\}+\backslash overline\{BD\})\}\{2\}\backslash cdot\backslash overline\{ED\}\backslash quad\backslash Rightarrow\backslash quad\; A\_\backslash text\{\{ABDE\}\}\backslash \; =\backslash \; \backslash dfrac\{(13\{,\}9\backslash \; dm+7\{,\}7\backslash \; dm)\}\{2\}\backslash cdot\; 22\backslash \; dm$

$\underset{‾}{\underset{‾}{A_{ABDE}\approx \mathrm{237,6}d{m}^2}}\backslash hspace\{9mm\}\backslash underline\{\backslash underline\{\{A\_\{ABDE\}\}\backslash \; \backslash approx\backslash \; 237\{,\}6\backslash \; dm\; ^2\}\}$

Der Kathetensatz besagt, dass jeweils das Quadrat einer Kathete gleich dem Produkt des anliegenden Achsenabschnitts der Hypotenuse und der Hypotenuse selbst ist.

$\underset{‾}{\text{Anwendung des Kathetensatzes auf die oben abgebildete Skizze}}\backslash underline\{\backslash text\{Anwendung\; des\; Kathetensatzes\; auf\; die\; oben\; abgebildete\; Skizze\}\}$

${\overline{CB}}^2=\overline{CF}\cdot \overline{CD}oder{\overline{BF}}^2=\overline{CF}\cdot \overline{DF}\backslash overline\{CB\}^2\backslash \; =\; \backslash overline\{CF\}\backslash cdot\backslash overline\{CD\}\backslash qquad\; oder\backslash qquad\; \backslash overline\{BF\}^2\backslash \; =\; \backslash overline\{CF\}\backslash cdot\backslash overline\{DF\}$