Konstruktion - Beweis 1

[size=200][i]Begründe  bzw. widerlege folgende Aussage:[br][br]In einem rechtwinkligen Dreieck mit einem 30[sup]0[/sup]-Winkel ist die kürzere Kathete[br]halb so lang wie die Hypotenuse.[br]Konstruiere dazu zunächst ein Dreieck ABC mit [br][math]\alpha[/math]= 30[sup]0[/sup] und [math]\gamma[/math] = 90[sup]0[/sup][br][br](Für die Begründung benötigst Du eine Hilfslinie!)[br][br][img]data:image/png;base64,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[/img][/i][/size]
Die benötigte Hilfslinie ist [MC].[br]ß = 90[sup]0[/sup] - a = 60[sup]0[/sup][br]Das Dreieck MBC ist daher gleichschenklig mit gleich großen Basiswinkeln bei M und C und ß = 60[sup]0[/sup].  Þ  Das Dreieck MBC ist sogar gleichseitig. 

Information: Konstruktion - Beweis 1