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Special Quadrilateral: Theorem 3

Special Quadrilateral: Theorem 3
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2017-02-10 05:39
PolygonCoordinate GeometryTransformation
In collections Congruence (CO) Cluster Congruence (Vol 2) CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Gail LeBlancCCSS High School: Geometry (Congruence) Volume 2 Unit 7: Polygons and Quadrilaterals Quadrilaterals
This example demonstrates Theorem 3 of special quadrilaterals, involving parallelograms, coordinates, means, right triangles, and quadratic equations. It is constructed using GeoGebra commands such as Rotate, Angle, Ray, Line, Point, Intersect, Segment, Min, Circle, Dilate, Vector, Polygon, Translate, Midpoint, CircleSector. Suitable for students at the junior high school level and above for learning geometric construction and related theorems.

GeoGebra Commands & Parameters

  1. Rotate(B, a°, A) → B'
  2. Angle(B, A, B') → α
  3. Ray(A, B) → f
  4. Ray(A, B') → g
  5. Line(B, g) → h
  6. Point(g) → C
  7. Line(C, f) → i
  8. Intersect(h, i) → D
  9. Segment(C, D) → j
  10. Segment(A, B) → k
  11. Segment(B, D) → l
  12. Segment(A, C) → m
  13. Min(m, k) → b
  14. Circle(C, (0.2 * b)) → d
  15. Circle(A, (0.2 * b)) → e
  16. Circle(B, (0.2 * b)) → p
  17. Circle(D, (0.2 * b)) → c
  18. Intersect(e, k, 1) → E
  19. Intersect(e, m, 1) → F
  20. Intersect(p, k, 1) → G
  21. Intersect(p, l, 1) → H
  22. Intersect(d, g, 2) → I
  23. Ray(C, I) → q
  24. Intersect(c, h, 2) → J
  25. Ray(D, J) → r
  26. Intersect(c, i, 2) → K
  27. Ray(D, K) → s
  28. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], A) → A'
  29. Rotate(E, If[1 < o ≤ 2, ((o - 1) * α), α], A) → E'
  30. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], C) → A'_1
  31. Vector(A, A'_1) → u
  32. Segment(A, A'_1) → f_1
  33. Dilate(C, If[2 < o ≤ 3, 3 - o, 0], D) → C'
  34. Vector(C, C') → v
  35. Segment(C, C') → g_1
  36. Dilate(D, If[3 < o ≤ 4, 4 - o, 0], B) → D'
  37. Segment(D, D') → h_1
  38. Polygon(A, B, D, C) → poly1, a_1, b_1, d_2, c_2
  39. Dilate(A, If[5.5 < o ≤ 6.5, 6.5 - o, 0], C) → A'_2
  40. Vector(A, A'_2) → w
  41. Translate(A, w) → A'_3
  42. Translate(A', w) → A''
  43. Dilate(A, If[6.5 < o ≤ 7.5, 7.5 - o, 0], B) → A'_4
  44. Vector(A, A'_4) → u_1
  45. Translate(A, u_1) → A'_5
  46. Translate(A'_1, u_1) → L
  47. Intersect(d, i) → M, N
  48. Intersect(d, g) → O, P
  49. Angle(A, C, N) → β
  50. Rotate(N, If[6.5 < o ≤ 7.5, ((-(o - 6.5)) * β), (-β)], C) → N'
  51. Dilate(C, If[7.5 < o ≤ 8.5, 8.5 - o, 0], B) → C'_1
  52. Vector(C, C'_1) → v_1
  53. Midpoint(A, D) → Q
  54. Dilate(A, If[4.5 < o ≤ 5.5, 5.5 - o, 0], Q) → A'_6
  55. Segment(A, A'_6) → i_1
  56. Dilate(Q, If[5.5 < o ≤ 6.5, 6.5 - o, 0], D) → Q'
  57. Segment(Q, Q') → j_1
  58. Dilate(B, If[6.5 < o ≤ 7.5, 7.5 - o, 0], Q) → B'_1
  59. Segment(B, B'_1) → k_1
  60. Dilate(Q, If[7.5 < o ≤ 8.5, 8.5 - o, 0], C) → Q'_1
  61. Segment(Q, Q'_1) → l_1
  62. Rotate(A, If[8.5 < o ≤ 9.5, ((-(o - 8.5)) * π), (-π)], Q) → A'_7
  63. Rotate(A'_6, If[9.5 < o ≤ 10.5, ((-(o - 9.5)) * π), (-π)], Q) → R
  64. Segment(A'_7, R) → m_1
  65. Rotate(B, If[9.5 < o ≤ 10.5, ((-(o - 9.5)) * π), (-π)], Q) → B'_2
  66. Rotate(B'_1, If[10.5 < o ≤ 11.5, ((-(o - 10.5)) * π), (-π)], Q) → S
  67. Segment(B'_2, S) → n_1
  68. Line(A, B) → p_1
  69. Intersect(e, p_1) → T, U
  70. Ray(A, T) → q_1
  71. Intersect(e, f_1) → V,
  72. Intersect(d, i, 1) → W
  73. Ray(C, W) → r_1
  74. Segment(B, A') → t
  75. Rotate(V, If[11 < o ≤ 12, ((o - 11) * (π - α)), π - α], A) → V'
  76. CircleSector(A, V, V') → c_1
  77. Dilate(A, If[12 < o ≤ 13, 13 - o, 0], C) → A'_8
  78. Vector(A, A'_8) → w_1
  79. Translate(c_1, w_1) → d_1
  80. Dilate(C, If[13 < o ≤ 14, 14 - o, 0], D) → C'_2
  81. Vector(C, C'_2) → e_1
  82. Translate(d_1, e_1) → s_1

AI Drawing Prompt Reference

Draw a parallelogram ABCD with points A, B, C, and D as vertices. First, rotate point B around point A by α degrees to get B'. Then, draw ray g through A and B', and draw line h through B. The intersection of g and h is point C. Next, draw line i through C, and the intersection of i and h is point D. Connect C and D, A and B, B and D, and A and C. Then, draw circle d with center C and radius 0.2 times AC, circle e with center A and radius 0.2 times AB, circle p with center B and radius 0.2 times BC, and circle c with center D and radius 0.2 times CD. The intersection of e and k is point E, the intersection of e and m is point F, the intersection of p and k is point G, and the intersection of p and l is point H. The intersection of d and g is point I. Then, draw ray q from C, and the intersection of h and i is point J. Draw ray r from D, and the intersection of i and j is point K. Draw ray s from D. Next, translate A along the vector from A to A' to get A', translate E along the vector from E to E' to get E', translate A along the vector from A to A'_1 to get A'_1, translate C along the vector from C to C' to get C', translate D along the vector from D to D' to get D', translate A along the vector from A to A'_2 to get A'_2, translate A along the vector from A to A'_3 to get A'_3, translate A' along the vector from A' to A'' to get A'', translate A along the vector from A to A'_4 to get A'_4, translate A along the vector from A to A'_5 to get A'_5, translate L along the vector from L to M to get M, translate L along the vector from L to N to get N, translate L along the vector from L to O to get O, and translate L along the vector from L to P to get P.
The prompt above was compiled from the original drawing commands and is for reference only.