Generalization through problem solving

1. 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

2. 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

3. The tangram Part II / 3 – Cut a quadrilateralinto 2 halves Gergely Wintsche

4. The pentominos Part II / 4 – Cut a quadrilateralinto 2 halves GergelyWintsche

5. 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

6. 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

7. Introduction The Wallace-Bolyai-Gerwientheorem The moving version: Part II / 7 – Cut a quadrilateralinto 2 halves Gergely Wintsche

8. Introduction The Wallace-Bolyai-Gerwientheorem 3. Anyrectangle is dissectedinto a rectanglewithagivenside. Part II / 8 – Cut a quadrilateralinto 2 halves Gergely Wintsche

9. 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

10. Introduction The basicproblem LetusprovethatthetAED (red) and thetBCE(green ) areasareequal. Part II / 10 – Cut a quadrilateralinto 2 halves Gergely Wintsche

11. 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

12. 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

13. 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

14. Introduction Quadrilateral CuttheABCD quadrilateralintotwohalveswith a line thatgoesthroughthemidpoint of theAD edge. Part II / 14 – Cut a quadrilateralinto 2 halves Gergely Wintsche

15. Introduction Solution (1) CuttheABCD quadrilateralintotwohalveswith a line thatgoesthroughthemidpoint of theAD edge. Part II/ 15 – Cut a quadrilateralinto 2 halves Gergely Wintsche

16. Introduction Solution (2) CuttheABCD quadrilateralintotwohalveswith a line thatgoesthroughthemidpoint of theAD edge. Part I / 16 – Cut a quadrilateralinto 2 halves Gergely Wintsche

17. Introduction The quadrilateral CuttheABCD quadrilateralintotwohalveswith a line thatgoesthroughthePpointontheedges. Part II / 17 – Cut a quadrilateralinto 2 halves Gergely Wintsche