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Parallelogram Folding: Exploring Reflection and Symmetry

Parallelogram Folding: Exploring Reflection and Symmetry
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2018-02-26 17:44
PolygonAngleTransformationGeometric ConstructionLocus
In collections Folding paper, doing geometry figures
① This GeoGebra example dynamically demonstrates the folding of a parallelogram along a diagonal or crease, visually showing the position and shape changes before and after the fold. ② Core mathematical concepts include properties of parallelograms, reflection (axial symmetry) transformations, congruent figures, midpoints and perpendicular lines, as well as congruence and area relationships among the resulting triangles and polygons. ③ Construction highlights: using Polygon to create the parallelogram, Midpoint for midpoints, CircleArc for auxiliary arcs, Rotate for the folding rotation, Intersect for intersection points, OrthogonalLine for perpendiculars, and Line/Point to define creases and key points; the object list includes multiple vertices, edges, triangles, and rotated copies (e.g., B', E', D'), illustrating dynamic construction and transformation. ④ Suitable for middle and high school students learning geometric transformations, congruence, and symmetry, and for teachers to demonstrate folding problems dynamically in class, helping students understand the mathematical essence of folding.

GeoGebra Commands & Parameters

  1. Polygon(A, C, D, B) → vierhoek1, a, c, d, b
  2. Midpoint(B, D) → BD
  3. Midpoint(A, C) → F
  4. Midpoint(B, E) → G
  5. CircleArc(B, DD, D) → e
  6. Segment(B, E) → f
  7. Polygon(B, A, C, E) → vierhoek2, b_1, a_1, c_1, e_1
  8. Polygon(B, E, D) → driehoek1, d_1, b_2, e_2
  9. Rotate(B, (((-(s - 1)) * 180))°, e_1) → B'
  10. Rotate(E, (((-(s - 1)) * 180))°, e_1) → E'
  11. Rotate(D, (((-(s - 1)) * 180))°, e_1) → D'
  12. Polygon(B', E', D') → driehoek1', d', b', e'
  13. Polygon(DD, E, B) → driehoek2, b_3, dd, e_3
  14. Segment(G, F) → g
  15. Polygon(G, F, C, E) → vierhoek3, g_1, f_1, c_2, e_4
  16. Polygon(B, A, F, G) → vierhoek4, b_4, a_2, f_2, g_2
  17. Rotate(G, (((-(s - 3)) * 180))°, f_2) → G'
  18. Rotate(F, (((-(s - 3)) * 180))°, f_2) → F'
  19. Rotate(C, (((-(s - 3)) * 180))°, f_2) → C'
  20. Rotate(E, (((-(s - 3)) * 180))°, f_2) → E'_1
  21. Polygon(G', F', C', E'_1) → vierhoek3', g', f', c', e'_1
  22. Polygon(A, F, G, DD) → vierhoek5, a_3, f_3, g_3, dd_1
  23. Segment(FE, F) → i
  24. Polygon(FE, A, F) → driehoek3, f_4, fe, a_4
  25. Rotate(FE, (((-(s - 5)) * 180))°, i) → FE'
  26. Rotate(A, (((-(s - 5)) * 180))°, i) → A'
  27. Rotate(F, (((-(s - 5)) * 180))°, i) → F'_1
  28. Polygon(FE', A', F'_1) → driehoek3', f'_1, fe', a'
  29. Polygon(B, FE, F, G) → vierhoek6, b_5, fe_1, f_5, g_4
  30. Intersect(g_3, i) → H
  31. OrthogonalLine(H, g', xOyPlane) → j
  32. Line(BD, F) → n
  33. Point(n) → K
  34. CircleArc(K, C, A) → p
  35. Polygon(G, H, F) → driehoek4, f_6, g_5, h_1
  36. Intersect(f_6, Line[FE, GG]) → J
  37. OrthogonalLine(H, b) → k
  38. Intersect(f_5, k) → I
  39. Polygon(G, H, I) → driehoek5, i_1, g_6, h_2
  40. Polygon(FE, F, L) → driehoek6, l, fe_2, f_7
  41. Intersect(f_6, f_7) → M
  42. Segment(G, M) → m
  43. Segment(H, M) → q
  44. CircleArc(F, FA, A) → h

AI Drawing Prompt Reference

Draw a parallelogram and demonstrate its folding along a diagonal or crease. Start with the parallelogram, mark the midpoints of both diagonals, and connect these midpoints with auxiliary lines. Rotate one part of the parallelogram about a diagonal to show the folded triangle and quadrilateral, illustrating the change in position. Then rotate another section about a different crease to create overlapping triangles and quadrilaterals. Finally, add auxiliary arcs, perpendicular lines, and intersection points to emphasize symmetry, congruence, and area relationships, presenting a clear geometric transformation.
The prompt above was compiled from the original drawing commands and is for reference only.