Generalization through problem solving

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Generalization through problem solving. Part II. The Wallace-Bolyai-Gerwien theorem Cut a quadrilateral into 2 halves. Gergely Wintsche Mathematics Teaching and Didactic Center Faculty of Science Eötvös Loránd University, Budapest.

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Generalization

throughproblemsolving

Part II.

The Wallace-Bolyai-Gerwientheorem

Gergely Wintsche

Mathematics Teaching and Didactic Center

Faculty of Science

Eötvös Loránd University, Budapest

CME12, 2012.07.02.– Rzeszów, Poland Gergely Wintsche

Outline

• 1. Dissections, examples
• 2. The Wallace-Bolyai-Gerweintheorem
• The basic lemma
• Triangle
• Trapezoid

Part II / 2 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

The tangram

Part II / 3 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

The pentominos

Part II / 4 – Cut a quadrilateralinto 2 halves

GergelyWintsche

Introduction

The Wallace-Bolyai-Gerwientheorem

„Twofiguresarecongruentbydissectionwheneithercan be dividedintopartswhicharerespectivelycongruentwiththecorrespondingpartstheother.” (Wallace)

Any two simple polygons of equal area can be dissected into a finite number of congruent polygonal pieces.

Part II / 5 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

The Wallace-Bolyai-Gerwientheorem

Letusdoitbysteps:

Proovethatanytriangle is dissectedinto a parallelogramma.

Any parallelogramma is dissectedinto a rectangle.

Part II / 6 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

The Wallace-Bolyai-Gerwientheorem

The moving version:

Part II / 7 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

The Wallace-Bolyai-Gerwientheorem

3. Anyrectangle is dissectedinto a rectanglewithagivenside.

Part II / 8 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

The Wallace-Bolyai-Gerwientheorem

Letustriangulatethesimplepolygon.

Everytriangle is dissectedinto a rectangle.

Everyrectangle is dissectedintorectangleswith a sameside and all of themforms a bigrectangle.

Wecandothesamewiththeotherpolygonand wecantailorthetworectanglesintoeachother.

Part II / 9 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

The basicproblem

LetusprovethatthetAED (red) and thetBCE(green ) areasareequal.

Part II / 10 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

The triangle

There is a givenPpointontheACside of an ABCtriangle. Constract a line throughPwhichcutthearea of thetriangletwohalf.

Part II / 11 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

Construct a line throughthevertexA of theABCDconvexquadrilateralwhichcutsthearea of itintotwohalves.

(Varga Tamás Competition 89-90, grade 8.)

Part II / 12 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

The trapezoid

Construct a line throughthemidpoint of theAD, whichhalvesthearea of theABCD trapezoid.

(Kalmár László Competition 93, grade 8.)

Part II / 13 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

Part II / 14 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

Solution (1)

Part II/ 15 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

Solution (2)

Part I / 16 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

Introduction

Part II / 17 – Cut a quadrilateralinto 2 halves

Gergely Wintsche