Geometrie-Dreieck-Kongruenzsätze



Beispiel Nr: 19
$\text{Seite-Seite-Seite }\\ a=4\quad b=5\quad c=3\\ \text{Pythagoras: } b^2=a^2+c^2 \\ b=\sqrt{a^2+c^2} \\ b=\sqrt{4^2+3^2}\\ b=5 \text{ Rechtwinkliges Dreieck }\\ \text{Kathete: } \quad a=4\quad\text{Hypothenuse: }\quad b=5\quad \text{Kathete: } \quad c=3\quad \beta=90^\circ\\ \\ \text{Sinus: }\quad \sin\alpha= \displaystyle \frac{a}{b} \\ \sin\alpha= \displaystyle \frac{4}{5} \\ \alpha=53,1 \\ \text{Winkelsumme: } \alpha + \beta + \gamma =180^\circ\\ \alpha + \beta + \gamma =180 \qquad /-\alpha \qquad /-\beta \\ \gamma =180^\circ -\alpha -\beta \\ \gamma =180^\circ -53,1^\circ - 90^\circ \\ \gamma =36,9^\circ \\ \text{Umfang: } U=a+b+c \\ U=4+5+3 \\ U=12 \\ \text{Höhe: } h_a \\ \sin\beta= \displaystyle \frac{h_a}{c} \\ \sin\beta= \displaystyle \frac{h_a}{c} \quad /\cdot c\\ h_a =c \cdot \sin\beta \\ h_a =3 \cdot \sin90^\circ \\ h_a=3 \\ \text{Flaeche: } \quad A = \frac{1}{2}\cdot a \cdot h_a \\ A = \frac{1}{2}\cdot 4 \cdot 3 \\ A=6 \\ \text{Höhe: } h_b \\ \sin\gamma= \displaystyle \frac{h_b}{a} \\ \sin\gamma= \displaystyle \frac{h_b}{a} \quad /\cdot a\\ h_b =a \cdot \sin\gamma \\ h_b =4 \cdot \sin36,9^\circ \\ h_b=2\frac{2}{5} \\ \text{Höhe: } h_c \\ \sin\alpha= \displaystyle \frac{h_c}{b} \\ \sin\alpha= \displaystyle \frac{h_c}{b} \quad / \cdot b\\ h_c=b \cdot \sin\alpha \\ h_c=5 \cdot \sin53,1^\circ \\ h_c=4 \\ \text{Winkelhalbierende: }\alpha \\ \delta=180-\beta-\frac{\alpha}{2} \\ \text{Sinus-Satz:} \displaystyle \frac{wha}{\sin\beta}=\frac{c}{\sin\delta } \\ \displaystyle \frac{wha}{\sin \beta}=\frac{c}{\sin\delta }\qquad /\cdot \sin\beta \\ wha=\displaystyle\frac{c \cdot \sin\beta}{ \sin\delta } \\ wha =\displaystyle\frac{3\cdot \sin90 }{ \sin63,4} \\ wha=3,35 \\ \text{Winkelhalbierende: }\beta \\ \delta=180-\frac{\beta}{2}-\gamma \\ \text{Sinus-Satz:} \displaystyle \frac{whb}{\sin\gamma}=\frac{a}{\sin\delta } \\ \displaystyle \frac{whb}{\sin \gamma}=\frac{a}{\sin\delta }\qquad /\cdot \sin\gamma \\ whb=\displaystyle\frac{a \cdot \sin\gamma}{ \sin\delta } \\ whb =\displaystyle\frac{4\cdot \sin36,9 }{ \sin98,1} \\ whb=2,42 \\ \text{Winkelhalbierende: }\gamma \\ \delta=180-\alpha-\frac{\gamma}{2} \\ \text{Sinus-Satz:} \displaystyle \frac{whc}{\sin\alpha}=\frac{b}{\sin\delta } \\ \displaystyle \frac{whc}{\sin \alpha}=\frac{b}{\sin\delta }\qquad /\cdot \sin\alpha \\ whc=\displaystyle\frac{b \cdot \sin\alpha}{ \sin\delta } \\ whc =\displaystyle\frac{5\cdot \sin53,1 }{ \sin63,4} \\ whc=3,58 \\ \text{Seitenhalbierende: } \\ s_a=\frac{1}{2}\sqrt{2(b^2+c^2)-a^2} \\ s_a=\frac{1}{2}\sqrt{2(5^2+3^2)-4^2} \\ s_a=3,61 \\ \text{Seitenhalbierende: } s_b=\frac{1}{2}\sqrt{2(a^2+c^2)-b^2}\\ s_b=\frac{1}{2}\sqrt{2(4^2+3^2)-5^2}\\ s_b=2\frac{1}{2} \\ \text{Seitenhalbierende: } s_c=\frac{1}{2}\sqrt{2(a^2+b^2)-c^2}\\ s_c=\frac{1}{2}\sqrt{2(4^2+5^2)-3^2}\\ s_c=3,77 \\ \text{Umkreisradius: } 2\cdot r_u= \displaystyle \frac{a}{\sin\alpha} \\ r_u =\displaystyle\frac{a}{2\cdot\sin\alpha} \\ r_u =\displaystyle\frac{4}{2\cdot\sin53,1^\circ} \\ r_u=2\frac{1}{2} \\ \text{Inkreisradius: }r_i= \displaystyle \frac{2 \cdot A}{U} \\ r_i= \displaystyle \frac{2 \cdot 6}{12} \\ r_i=1 \\\\\psset{xunit=1.00cm,yunit=1.00cm} \begin{pspicture}*(-2,-2)(5.00,6.00) \pstTriangle[PointSymbol=none](0.00,0.00){A}(3.00,0.00){B}(3.00,4.00){C} \pstMarkAngle[linecolor=cyan]{B}{A}{C}{$\alpha$}\pstMarkAngle[linecolor=cyan]{C}{B}{A}{$\beta$}\pstMarkAngle[linecolor=cyan]{A}{C}{B}{$\gamma$ }\pstMiddleAB[PointSymbol=none]{A}{C}{b}\pstMiddleAB[PointSymbol=none]{C}{B}{a}\pstMiddleAB[PointSymbol=none,PosAngle=-100]{A}{B}{c} \end{pspicture} \\ \text{Seitenhalbierende-Schwerpunkt} \\\psset{xunit=1.00cm,yunit=1.00cm} \begin{pspicture}*(-2,-2)(5.00,6.00) \pstTriangle[PointSymbol=none](0.00,0.00){A}(3.00,0.00){B}(3.00,4.00){C} \psset{PointSymbol=none} { \psset{linestyle=none, PointName=none} \pstMediatorAB{A}{B}{K}{KP} \pstMediatorAB[PosAngleA=-40]{C}{A}{J}{JP} \pstMediatorAB[PosAngleA=75]{B}{C}{I}{IP} } \psset{linecolor=magenta}\ncline{A}{I}\ncline{C}{K}\ncline{B}{J} \pstMiddleAB[PointSymbol=none,PosAngle=-10]{A}{I}{s_a} \pstMiddleAB[PointSymbol=none,PosAngle=-10]{C}{K}{s_c} \pstMiddleAB[PointSymbol=none,PosAngle=-10]{B}{J}{s_b} \pstCGravABC[PointSymbol=square, PosAngle=5]{A}{B}{C}{M} \end{pspicture} \\ \text{Mittelsenkrechte - Umkreis} \\ \psset{xunit=1.00cm,yunit=1.00cm} \begin{pspicture}*(-2,-2)(5.00,6.00) \pstTriangle[PointSymbol=none](0.00,0.00){A}(3.00,0.00){B}(3.00,4.00){C} \psset{PointSymbol=none} {\psset{linestyle=none, PointName=none} \pstMediatorAB{A}{B}{K}{KP} \pstMediatorAB[PosAngleA=-40]{C}{A}{J}{JP} \pstMediatorAB[PosAngleA=75]{B}{C}{I}{IP} } \pstInterLL[PointSymbol=square, PosAngle=-170]{I}{IP}{J}{JP}{O} \pstMiddleAB[PointSymbol=none,PosAngle=-10]{K}{KP}{m_c} \pstMiddleAB[PointSymbol=none,PosAngle=-10]{J}{JP}{m_b} \pstMiddleAB[PointSymbol=none,PosAngle=-10]{I}{IP}{m_a} { \psset{nodesep=-.8, linecolor=green} \pstLineAB{O}{I}\pstLineAB{O}{J}\pstLineAB{O}{K} } \pstCircleOA[linecolor=red]{O}{A} \end{pspicture} \\\text{Höhen} \\ \psset{xunit=1.00cm,yunit=1.00cm} \begin{pspicture}*(-2,-2)(5.00,6.00) \pstTriangle[PointSymbol=none](0.00,0.00){A}(3.00,0.00){B}(3.00,4.00){C} \psdot[dotstyle=square](O) {\psset{ PointName=none} \pstProjection{B}{A}{C} \pstProjection{B}{C}{A} \pstProjection{A}{C}{B} } \psset{linecolor=blue}\ncline{A}{A'}\ncline{C}{C'}\ncline{B}{B'} \pstInterLL[PointSymbol=square]{A}{A'}{B}{B'}{H} \pstMiddleAB[PointSymbol=none,PosAngle=-10]{A}{A'}{h_a} \pstMiddleAB[PointSymbol=none,PosAngle=-10]{C}{C'}{h_c} \pstMiddleAB[PointSymbol=none,PosAngle=-10]{B}{B'}{h_b} \end{pspicture} \\ \text{Winkelhalbierende-Inkreis} \\ \psset{xunit=1.00cm,yunit=1.00cm} \begin{pspicture}*(-2,-2)(5.00,6.00) \pstTriangle[PointSymbol=none](0.00,0.00){A}(3.00,0.00){B}(3.00,4.00){C} {\psset{linestyle=none, PointName=none,PointSymbol=none} \pstBissectBAC{C}{B}{A}{B'} \pstBissectBAC{B}{A}{C}{A'} \pstBissectBAC{A}{C}{B}{C'} \pstInterLL{A}{B}{C}{C'}{E} \pstInterLL{A}{C}{B}{B'}{F} \pstInterLL{B}{C}{A}{A'}{G} } {\psset{linecolor=blue} \ncline{A}{G}\ncline{B}{F}\ncline{C}{E} \pstInterLL[PointSymbol=square]{A}{A'}{B}{B'}{M_i} \pstMiddleAB[PointSymbol=none,PosAngle=1]{A}{A'}{w_a} \pstMiddleAB[PointSymbol=none,PosAngle=1]{C}{C'}{w_c} \pstMiddleAB[PointSymbol=none,PosAngle=1]{B}{B'}{w_b} } \pstProjection[PointSymbol=none]{B}{A}{M_i} \pstCircleOA[linecolor=red]{M_i}{M_i'} \end{pspicture} \\ $