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Coordinate Geometry Proof: Diagonals of a Rectangle are Congruent (1)

Coordinate Geometry Proof: Diagonals of a Rectangle are Congruent (1)
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2018-12-04 09:47
TrianglePolygonCoordinate GeometryTransformationGeometric Construction
① This GeoGebra example demonstrates and proves that the diagonals of a rectangle are congruent using coordinate geometry. ② Core mathematical concepts include rectangle properties, diagonals, congruent triangles, distance formula, and formal geometric proof. ③ Construction highlights: based on the summary, it uses commands such as Segment, Line, Ray, OrthogonalLine, Intersect, Polygon, Slope, Translate, Rotate, Dilate, Angle, and Circle to build the rectangle and its diagonals, enabling dynamic verification. ④ Suitable for middle or high school geometry and coordinate geometry lessons, ideal for interactive proof demonstrations and helping students grasp the power of analytic methods in geometry.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Slope(a) → c
  3. OrthogonalLine(A, a) → e
  4. Point(e) → C
  5. Ray(A, C) → d
  6. Point(d) → D
  7. Segment(A, D) → f
  8. Line(D, a) → g
  9. Line(B, f) → h
  10. Intersect(g, h) → E
  11. Segment(D, E) → i
  12. Segment(E, B) → j
  13. Circle(A, (0.2 * Min[f, i])) → k
  14. Intersect(k, f) → G,
  15. Intersect(k, a, 1) → H
  16. Dilate(A, If[0 < l ≤ 1, 1 - l, 0], B) → A'
  17. Vector(A, A') → u
  18. CircleSector(A, H, G) → p
  19. Translate(p, u) → p'
  20. Rotate(p', If[1 < l ≤ 2, ((l - 1) * π / 2), 1.5707963267948966], B) → p''
  21. Dilate(B, If[2 < l ≤ 3, 3 - l, 0], E) → B'
  22. Vector(B, B') → v
  23. Translate(p'', v) → p'''
  24. Rotate(p''', If[3 < l ≤ 4, ((l - 3) * π / 2), 1.5707963267948966], E) → q
  25. Dilate(E, If[4 < l ≤ 5, 5 - l, 0], D) → E'
  26. Vector(E, E') → w
  27. Translate(q, w) → q'
  28. Rotate(q', If[5 < l ≤ 6, ((l - 5) * π / 2), 1.5707963267948966], D) → q''
  29. Dilate(D, If[6 < l ≤ 7, 7 - l, 0], B) → D'
  30. Vector(D, D') → m
  31. Segment(D, D') → n
  32. Dilate(A, If[7 < l ≤ 8, 8 - l, 0], E) → A'_1
  33. Segment(A, A'_1) → s
  34. Polygon(A, B, E, D) → poly1, a_1, b_1, e_1, d_1
  35. Dilate(A, If[16 < l ≤ 17, 17 - l, 0], E) → A'_2
  36. Dilate(A, If[16 < l ≤ 17, 17 - l, 0], B) → A'_3
  37. Dilate(A, If[16 < l ≤ 17, 17 - l, 0], D) → A'_4
  38. Vector(A, A'_3) → u_1
  39. Vector(A, A'_2) → v_1
  40. Vector(A, A'_4) → w_1
  41. Dilate(A, If[18 < l ≤ 19, 19 - l, 0], E) → A'_5
  42. Dilate(A, If[18 < l ≤ 19, 19 - l, 0], D) → A'_6
  43. Dilate(A, If[18 < l ≤ 19, 19 - l, 0], B) → A'_7
  44. Vector(A, A'_7) → i_1
  45. Vector(A, A'_5) → j_1
  46. Vector(A, A'_6) → l_1
  47. Dilate(D, If[8 < l < 9, 9 - l, 0], E) → D'_1
  48. Vector(D, D'_1) → r
  49. Translate(D, r) → D'_2
  50. Translate(D', r) → D''
  51. Segment(D'_2, D'') → n'
  52. Angle(A, D, B) → β
  53. Rotate(D'_2, If[9 < l ≤ 10, ((-2 * (l - 9)) * β), (-2 * β)], E) → L
  54. Rotate(D'', If[9 < l ≤ 10, ((-2 * (l - 9)) * β), (-2 * β)], E) → D'''
  55. Segment(L, D''') → n''