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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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slide1

Generalization

throughproblemsolving

Part II.

The Wallace-Bolyai-Gerwientheorem

Cut a quadrilateralinto 2 halves

Gergely Wintsche

Mathematics Teaching and Didactic Center

Faculty of Science

Eötvös Loránd University, Budapest

CME12, 2012.07.02.– Rzeszów, Poland Gergely Wintsche

slide2

Outline

  • 1. Dissections, examples
  • 2. The Wallace-Bolyai-Gerweintheorem
  • 3. Cutting a quadrilateral
  • The basic lemma
  • Triangle
  • Trapezoid
  • Quadrilateral

Part II / 2 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

slide3

The tangram

Part II / 3 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

slide4

The pentominos

Part II / 4 – Cut a quadrilateralinto 2 halves

GergelyWintsche

slide5

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

slide6

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

slide7

Introduction

The Wallace-Bolyai-Gerwientheorem

The moving version:

Part II / 7 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

slide8

Introduction

The Wallace-Bolyai-Gerwientheorem

3. Anyrectangle is dissectedinto a rectanglewithagivenside.

Part II / 8 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

slide9

Introduction

The Wallace-Bolyai-Gerwientheorem

Weareready!

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

slide10

Introduction

The basicproblem

LetusprovethatthetAED (red) and thetBCE(green ) areasareequal.

Part II / 10 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

slide11

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

slide12

Introduction

The quadrilateral

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

slide13

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

slide14

Introduction

Quadrilateral

CuttheABCD quadrilateralintotwohalveswith a line thatgoesthroughthemidpoint of theAD edge.

Part II / 14 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

slide15

Introduction

Solution (1)

CuttheABCD quadrilateralintotwohalveswith a line thatgoesthroughthemidpoint of theAD edge.

Part II/ 15 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

slide16

Introduction

Solution (2)

CuttheABCD quadrilateralintotwohalveswith a line thatgoesthroughthemidpoint of theAD edge.

Part I / 16 – Cut a quadrilateralinto 2 halves

Gergely Wintsche

slide17

Introduction

The quadrilateral

CuttheABCD quadrilateralintotwohalveswith a line thatgoesthroughthePpointontheedges.

Part II / 17 – Cut a quadrilateralinto 2 halves

Gergely Wintsche