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Änderungen von Dokument Lösung Problemlösen

Zuletzt geändert von Martin Rathgeb am 2025/11/17 00:55
Von Version 1.1
bearbeitet von Nicole Böhringer
am 2025/11/06 13:45
Änderungskommentar: Es gibt keinen Kommentar für diese Version
Auf Version 7.1
bearbeitet von Martin Rathgeb
am 2025/11/17 00:52
Änderungskommentar: Es gibt keinen Kommentar für diese Version

Zusammenfassung

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Dokument-Autor
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1 -XWiki.bgr
1 +XWiki.martinrathgeb
Inhalt
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1 -Schritt 1:
1 +Vorbemerkung:
2 2  Die jeweils längsten Seiten der Figuren sind gleich groß. Dies wird hergeleitet über rechtwinklige Hilfsdreiecke und dem Kongruenzsatz für Dreiecke sws:
3 -[[image:geogebra-export_A4 Bild 3_Lösung1||width="500" style="display:block;margin-left:auto;margin-right:auto"]]
3 +[[image:geogebra-export_A4 Bild 3_Lösung1.svg||width="500" style="display:block;margin-left:auto;margin-right:auto"]]
4 +
5 +(%class=abc%)
6 +1. (((Zur Untersuchung der beiden Vierecke 8a und 8b betrachtet man ihre Struktur im Raster. Eine geeignete Methode besteht darin, jedes Viereck durch eine seiner Diagonalen in zwei Dreiecke zu zerlegen. Verbindet man in beiden Figuren dieselben beiden gegenüberliegenden Ecken, so entstehen in 8a und in 8b zwei Dreiecke, deren Seitenlängen und Winkel – anhand der Rasterpunkte – übereinstimmen. Die Dreiecke sind daher paarweise kongruent.
7 +
8 +Wählt man die zweite mögliche Diagonale, so erhält man erneut zwei Dreiecke, die in 8a und 8b jeweils dieselben Seiten- und Winkelverhältnisse besitzen. Auch diese Zerlegung zeigt, dass die entsprechenden Dreiecke kongruent sind.
9 +
10 +Da beide Vierecke bei beiden möglichen Zerlegungen in dieselben Paare kongruenter Dreiecke zerfallen und die Anordnung dieser Teilfiguren übereinstimmt, folgt, dass die Vierecke 8a und 8b kongruent sind. Sie besitzen dieselbe Form und Größe, unterscheiden sich jedoch in ihrer Lage im Raster.
11 +)))
12 +1. Um ein Viereck zu konstruieren, das nicht zu 8a und 8b kongruent ist, verwendet man die zwei Dreiecke, die bei der Zerlegung der ursprünglichen Vierecke auftreten. Man zeichnet zunächst das erste dieser Dreiecke wie in 8a oder 8b. Das zweite Dreieck wird kongruent dazu gezeichnet, jedoch in einer anderen Lage: beispielsweise gedreht oder gespiegelt oder an einer anderen gemeinsamen Seite angesetzt. Werden diese beiden Dreiecke so zusammengefügt, dass sie ein Viereck bilden, so entsteht eine Figur, die dieselben Teil-Dreiecke wie 8a und 8b enthält, aber insgesamt eine andere äußere Form besitzt.
13 +
14 +Die Nicht-Kongruenz lässt sich daran erkennen, dass die Lagebeziehung der beiden Dreiecke in der neuen Figur nicht derjenigen in 8a und 8b entspricht. Obwohl eine der Diagonalen wieder die beiden bekannten Dreiecke liefert, entsteht bei der anderen Diagonale ein Dreieckspaar,
15 +das nicht zu den Zerlegungsdreiecken von 8a und 8b kongruent ist. Damit ist das neue Viereck
16 +nicht kongruent zu den beiden gegebenen Vierecken, obwohl es aus denselben Dreiecken
17 +zusammengesetzt wurde.
18 +
geogebra-export_A4 Bild 3_Lösung1.ggb
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geogebra-export_A4 Bild 3_Lösung1.svg
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