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Rhombus Diagonals by Tim Brzezinski

Rhombus Diagonals by Tim Brzezinski
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2016-05-27 11:11
AngleCoordinate Geometry
In collections Expressing Geometric Properties with Equations (GPE) Cluster Unit 7: Polygons and Quadrilaterals Quadrilaterals Coordinate Geometry Proof Prompts
This example uses coordinate geometry to prove that the diagonals of a rhombus are perpendicular. It involves concepts such as rhombus, triangles, and ratios. Using GeoGebra commands like Circle, Line, Intersect, CircleArc, Point, Segment, Min, Dilate, Angle, Rotate, Polygon, Vector, CircleSector, it constructs a rhombus and verifies its diagonal properties. Suitable for middle school and above students learning geometric proofs.

GeoGebra Commands & Parameters

  1. Circle(B, A) → f
  2. Circle(A, B) → c
  3. Line(A, B) → b
  4. Intersect(c, b, 2) → C
  5. CircleArc(A, B, C) → d
  6. Point(d) → D
  7. Segment(D, B) → s
  8. Circle(D, A) → e
  9. Intersect(e, f, 2) → E
  10. Segment(A, E) → n
  11. Intersect(n, s) → L
  12. Segment(A, B) → a
  13. Segment(A, D) → g
  14. Segment(D, E) → h
  15. Segment(E, B) → i
  16. Min(g, h) → j
  17. Dilate(B, If[0 < l ≤ 1, 1 - l, 0], A) → B'
  18. Segment(B, B') → m
  19. Angle(E, B, A) → α
  20. Angle(D, E, B) → β
  21. Rotate(B, If[1 < l ≤ 2, ((-(l - 1)) * α), α], B) → B'_1
  22. Rotate(B', If[1 < l ≤ 2, ((-(l - 1)) * α), (-α)], B) → B''
  23. Segment(B'_1, B'') → m'
  24. Rotate(B'_1, If[2 < l ≤ 3, ((-(l - 2)) * β), (-β)], E) → J
  25. Rotate(B'', If[2 < l ≤ 3, ((-(l - 2)) * β), (-β)], E) → B'''
  26. Segment(J, B''') → m''
  27. Rotate(J, If[3 < l ≤ 4, ((-(l - 3)) * α), (-α)], D) → J'
  28. Rotate(B''', If[3 < l ≤ 4, ((-(l - 3)) * α), (-α)], D) → K
  29. Segment(J', K) → m'''
  30. Circle(B, (0.2 * j)) → k
  31. Circle(E, (0.2 * j)) → p
  32. Intersect(p, m'', 1) → F
  33. Intersect(p, m', 1) → G
  34. Intersect(k, m', 1) → H
  35. Intersect(k, m, 1) → I
  36. Polygon(A, B, E, D) → poly1, a_1, b_1, e_1, d_1
  37. Dilate(B, If[5 < l ≤ 6, 6 - l, 0], D) → B'_2
  38. Vector(B, B'_2) → u
  39. Dilate(E, If[7 < l ≤ 8, 8 - l, 0], A) → E'
  40. Vector(E, E') → v
  41. Dilate(D, If[4 < l ≤ 5, 5 - l, 0], B) → D'
  42. Vector(D, D') → w
  43. Min(n / 2, s / 2) → o
  44. Circle(L, (0.2 * o)) → t
  45. Intersect(t, s, 2) → M
  46. Intersect(t, n, 2) → N
  47. CircleSector(L, M, N) → c_1
  48. Segment(D, D') → f_1
  49. Dilate(L, If[10 < l ≤ 11, 11 - l, 0], B) → L'
  50. Circle(D, (0.2 * j)) → h_1
  51. Intersect(k, s) → P,
  52. Intersect(s, f_1) → Q
  53. Intersect(h_1, s, 1) → O
  54. Circle(A, (0.2 * j)) → k_1
  55. Intersect(k_1, m, 1) → R
  56. Intersect(k_1, n, 1) → S
  57. Intersect(k_1, m''', 1) → T
  58. Intersect(h_1, m'', 1) → U
  59. Dilate(A, If[5 < l ≤ 6, 6 - l, 0], E) → A'
  60. Segment(A, A') → i_1
  61. Dilate(L, If[12 < l ≤ 13, 13 - l, 0], E) → L'_1
  62. Intersect(p, m'', 1) → V
  63. Intersect(p, n, 1) → W
  64. Intersect(p, m', 1) → Z
  65. Rotate(L, If[13 < l ≤ 14, ((-(l - 13)) * π), -3.141592653589793], L) → L'_2
  66. Rotate(L', If[13 < l ≤ 14, ((-(l - 13)) * π), -3.141592653589793], L) → L''
  67. Rotate(L, If[14 < l ≤ 15, ((-(l - 14)) * π), 3.141592653589793], L) → L'_3
  68. Rotate(L'_1, If[14 < l ≤ 15, ((-(l - 14)) * π), 3.141592653589793], L) → A_1
  69. Rotate(c_1, If[6.25 < l ≤ 7.25, ((l - 6.25) * π / 2), 1.5708], L) → f_2