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Quadrilateral to Rectangle Dissection

Quadrilateral to Rectangle Dissection
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2015-09-09 15:17
PolygonCoordinate GeometryGeometric Construction
This example demonstrates how a quadrilateral can be dissected into a rectangle by drawing a line joining the mid-points of one pair of opposite sides. It involves core mathematical concepts like midpoints, segments, rotation, and intersection. The construction uses commands such as Polygon, Midpoint, Segment, Intersect, Point, Rotate, and Angle in GeoGebra. Suitable for high school students learning geometric constructions.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → poly1, a, b, c, d
  2. Midpoint(d) → F
  3. Midpoint(c) → G
  4. Midpoint(b) → H
  5. Segment(F, H) → e
  6. Intersect(OrthogonalLine[G, e], e) → J
  7. Point(h) → L
  8. Rotate(F, Angle[A, F, L], F) → F'
  9. Rotate(A, Angle[A, F, L], F) → A'
  10. Rotate(B, Angle[A, F, L], F) → B'
  11. Rotate(H, Angle[A, F, L], F) → H'_1
  12. Polygon(F', A', B', H'_1) → poly3, f', a', b', h'_1
  13. Midpoint(a') → E
  14. Intersect(OrthogonalLine[E, h'_1], h'_1) → I
  15. Point(f) → M
  16. Rotate(E, Angle[B', E, M], E) → E'
  17. Rotate(I, Angle[B', E, M], E) → I'
  18. Rotate(H'_1, Angle[B', E, M], E) → N
  19. Rotate(B', Angle[B', E, M], E) → B''
  20. Polygon(E', I', N, B'') → poly4, e', i', n, b''
  21. Polygon(L, E', I, F') → poly5, l, e'_1, i, f'_1
  22. Point(g) → K
  23. Rotate(F, Angle[D, F, K], F) → F'_1
  24. Rotate(H, Angle[D, F, K], F) → H'
  25. Rotate(C, Angle[D, F, K], F) → C'
  26. Rotate(D, Angle[D, F, K], F) → D'
  27. Polygon(F'_1, H', C', D') → poly2, f'_2, h', c', d'
  28. Rotate(J, Angle[D, F, K], F) → J'
  29. Rotate(G, Angle[D, F, K], F) → G'
  30. Point(p) → O
  31. Rotate(G', Angle[C', G', O], G') → G''
  32. Rotate(J', Angle[C', G', O], G') → J''
  33. Rotate(H', Angle[C', G', O], G') → H''
  34. Rotate(C', Angle[C', G', O], G') → C''
  35. Polygon(G'', J'', H'', C'') → poly6, g'', j'', h'', c''
  36. Polygon(K, F'_1, J', G') → poly7, k_1, f'_3, j', g'
  37. Segment(F, H) → m
  38. Segment(G, J) → q
  39. Segment(Midpoint[Segment[A, B]], Intersect[m, OrthogonalLine[Midpoint[Segment[A, B]], m]]) → r
  40. Angle(F, Intersect[m, OrthogonalLine[Midpoint[Segment[A, B]], m]], Midpoint[Segment[A, B]]) → α
  41. Angle(H, J, G) → β