Satz des Pythagoras: Erklärung

Den Satz des Pythagoras haben schon viele Lernende vor der eigentlichen Einführung während der Schuldzeit gehört. a²+b²=c² ist sicherlich einer der fundamentalen Sätze in der geometrischen Mathematik. Wozu dieser Satz genau dient und wie man ihn anwendet, erfährst du in diesem Artikel.

Als erstes möchte ich dir natürlich verraten wozu der Satz des Pythagoras dient! Mit Hilfe des Satzes kannst du eine fehlende Seitenlänge eines rechtwinkligen Dreiecks berechnen. Grundlage und somit zwingend erforderlich ist also ein rechtwinkliges Dreieck bei dem zwei Seiten bekannt sind.

Hier solltest du zunächst wissen, dass die beiden Seiten, die an dem rechten (also 90°) Winkel liegen, Katheten heißen. In diesem Beispiel sind es die Seiten a und b. Die gegenüberliegende Seite heißt Hypothenuse, hier c.

 

Nun kommen wir zu dem Satz des Pythagoras, den ich dir danach in leichteren, verständlicheren Worten erkläre:

„In allen rechtwinkligen Dreiecken ist die Summe der Flächeninhalte der Kathetenquadrate gleich dem Flächeninhalt des Hypotenusenquadrates.“

Erklärung: Die Katheten haben wir in unserem Beispiel mit a und b benannt. Die zugehörigen Quadrate sind also a² und b². Deren Summe (also Plus-Rechnung) ist genauso groß wie das Quadrat mit der Seitenlänge c, also c². Dies führt also zu a²+b²=c². Du weißt vielleicht, dass, wenn in einer Gleichung nur eine Unbekannte ist, dann kann man diese Unbekannte berechnen und genau deswegen kann man mit dieser Gleichung eine fehlende Seitenlänge eines rechtwinkligen Dreiecks ermitteln. Wenn man zum Beispiel die Länge der beiden Katheten kennt, die Länge der Hypothenuse fehlt, dann sind a und b bekannt und c kann berechnet werden.

Beispiel:

a = 5 cm    und    b = 3 cm    c = ?

In dem vorherigen Beispiel waren beide Katheten, also die Seiten am rechten Winkel, bekannt. Nun möchte ich dir noch ein zweites Beispiel zeigen, bei dem eine Kathete und die Hypothenuse bekannt ist.

Beispiel:

a = 6    und    c = 10     b = ?

Wie du siehst, löst du hier nach der unbekannten, zweiten Kathete auf und ziehst anschließend die Wurzel. Da der Satz des Pythagoras zur Berechnung von Seitenlängen an einem rechtwinkligen Dreieck dient, kann du die negative Lösung, die eine normale Quadratwurzel ja nun hat, weglassen, da eine Seitenlänge nicht negativ sein kann.

Erklärungsvideo auf Youtube:

 

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