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2 201 2 0 702 R z es zó w Poland Gergely Wintsche Generalization through problem solving Gergely Wintsche Mathematics Teaching and Didactic Center ID: 269041

halves quadrilateral gergely cut quadrilateral halves cut gergely part wintsche introduction wallace line bolyai abcd dissected theorem gerwien rectangle

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Slide1

CME1

2, 2012.07.02. – Rzeszów, Poland Gergely Wintsche

Generalization

through

problem solving

Gergely WintscheMathematics Teaching and Didactic CenterFaculty of ScienceEötvös Loránd University, Budapest

Part II.

The

Wallace-Bolyai-Gerwien

theorem

Cut

a

quadrilateral

into

2

halvesSlide2

Gergely

Wintsche

Outline

1.

Dissections

,

examples2. The Wallace-Bolyai-Gerwein theorem3. Cutting a

quadrilateralThe basic lemmaTriangleTrapezoid

Quadrilateral

Part

II

/

2

Cut

a

quadrilateral

into

2

halvesSlide3

Gergely

Wintsche

The

tangram

Part

II

/

3

Cut

a

quadrilateral

into

2

halvesSlide4

Gergely

Wintsche

The

pentominos

Part

II

/

4

Cut

a

quadrilateral

into

2

halvesSlide5

Gergely

Wintsche

Two

figures are

congruent by dissection

when either can be divided into parts which

are respectively congruent with the corresponding

parts

the

other

.”

(Wallace)

Part

II

/

5

Cut

a

quadrilateral

into

2

halves

Introduction

The

Wallace-Bolyai-Gerwien

theorem

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

.

Slide6

Gergely

Wintsche

Let

us do

it by steps:

Proove that any triangle is dissected into a parallelogramma.

Any parallelogramma is dissected into a rectangle.

Part

II

/

6

Cut

a

quadrilateral

into

2

halves

Introduction

The

Wallace-Bolyai-Gerwien

theoremSlide7

Gergely

Wintsche

The

moving

version:

Part

II

/

7

Cut

a

quadrilateral

into

2

halves

Introduction

The

Wallace-Bolyai-Gerwien

theoremSlide8

Gergely

Wintsche

3.

Any

rectangle is dissected into a

rectangle with a given side.

Part

II

/

8

Cut

a

quadrilateral

into

2

halves

Introduction

The

Wallace-Bolyai-Gerwien

theoremSlide9

Gergely

Wintsche

We

are

ready!Let us

triangulate the simple polygon. Every triangle is

dissected into a rectangle. Every rectangle is

dissected

into

rectangles

with

a

same

side

and

all

of

them

forms

a

big

rectangle

.

We

can

do

the

same

with

the other polygon and we can tailor the two rectangles

into each other.

Part II / 9 – Cut

a quadrilateral into 2 halves

Introduction

The

Wallace-Bolyai-Gerwien

theoremSlide10

Gergely

Wintsche

Let

us

prove that

the tAED (red) and the tBCE (

green ) areas are equal.

Part

II

/

10

Cut

a

quadrilateral

into

2

halves

Introduction

The

basic

problemSlide11

Gergely

Wintsche

There

is a

given

P point on

the AC side of an ABC triangle. Constract a line through

P which cut the area of the

triangle

two

half

.

Part

II

/

11

Cut

a

quadrilateral

into

2

halves

Introduction

The

triangleSlide12

Gergely

Wintsche

Construct

a line

through

the vertex A of

the ABCD convex quadrilateral which cuts the

area of it into two halves.

(Varga Tamás

Competition

89-90,

grade

8.)

Part

II

/

12

Cut

a

quadrilateral

into

2

halves

Introduction

The

quadrilateralSlide13

Gergely

Wintsche

Construct

a line

through

the midpoint of the

AD, which halves the area of the ABCD trapezoid.

(Kalmár László Competition 93, grade 8.)

Part

II

/

13

Cut

a

quadrilateral

into

2

halves

Introduction

The trapezoidSlide14

Gergely

Wintsche

Cut

the

ABCD quadrilateral into two

halves with a line that goes through the midpoint

of the AD edge.

Part

II

/

14

Cut

a

quadrilateral

into

2

halves

Introduction

QuadrilateralSlide15

Gergely

Wintsche

Cut

the

ABCD quadrilateral into two

halves with a line that goes through the midpoint

of the AD edge.

Part

II

/

15

Cut

a

quadrilateral

into

2

halves

Introduction

Solution

(1)Slide16

Gergely

Wintsche

Cut

the

ABCD quadrilateral into two

halves with a line that goes through the midpoint

of the AD edge.

Part I /

16

Cut

a

quadrilateral

into

2

halves

Introduction

Solution

(2)Slide17

Gergely

Wintsche

Cut

the

ABCD quadrilateral into two

halves with a line that goes through the P

point on the edges.

Part

II

/

17

Cut

a

quadrilateral

into

2

halves

Introduction

The

quadrilateral

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