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Coordinate Geometry Proof: Diagonals of a Rectangle are Congruent

Coordinate Geometry Proof: Diagonals of a Rectangle are Congruent
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2016-05-27 08:51
TrianglePolygonCoordinate GeometryGeometric Construction
In collections Expressing Geometric Properties with Equations (GPE) Cluster Unit 7: Polygons and Quadrilaterals Quadrilaterals Coordinate Geometry Proof Prompts
① This example demonstrates, through coordinate geometry, that the diagonals of a rectangle are congruent (equal in length). ② Core mathematical concepts include properties of rectangles, diagonal segments, congruent triangles, distance formula in coordinates, and logical proof structure. ③ Construction highlights: using GeoGebra commands such as Segment, Slope, OrthogonalLine, Point, Ray, Line, Intersect, Circle, Dilate, Vector, CircleSector, Translate, Rotate, Polygon, and Angle to build a rectangle and its diagonals, then measure and compare diagonal lengths to verify congruence. ④ Suitable for upper middle school or high school students learning coordinate geometry proofs, and useful for teachers as an interactive demonstration to illustrate rigorous geometric reasoning.

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''