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Relationship Among Segments Formed by Perpendiculars

Relationship Among Segments Formed by Perpendiculars
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2017-06-26 09:11
PolygonVectorGeometric Construction
This example illustrates the relationship among segments formed when a perpendicular is drawn to two of its sides in a quadrilateral. Key mathematical concepts include quadrilaterals, perpendicular segments, and similar triangles. It utilizes commands such as Rotate, Ray, Circle, Intersect, Line, Segment, OrthogonalLine, Angle, Dilate, Polygon, CircleSector, Point, Midpoint, Vector, and Translate in GeoGebra. Suitable for middle school and above students to learn about geometric constructions and similar triangles.

GeoGebra Commands & Parameters

  1. Rotate(B, (a * π / 180), A) → B'
  2. Ray(A, B') → j
  3. Circle(A, B) → c
  4. Intersect(c, j) → E,
  5. Line(B, E) → m
  6. Segment(A, B) → f
  7. Circle(B, f) → h
  8. Circle(E, f) → d
  9. Intersect(d, h, 1) → C
  10. Line(A, C) → l
  11. Intersect(l, m) → D
  12. Segment(E, C) → i
  13. OrthogonalLine(D, i) → n
  14. Circle(B, f) → e
  15. Angle(B, A, B') → α
  16. Segment(A, E) → g
  17. Segment(B, C) → k
  18. Intersect(n, f) → G
  19. Angle(D, G, A) → β
  20. OrthogonalLine(D, k) → q
  21. OrthogonalLine(D, g) → s
  22. Segment(D, A) → b
  23. Dilate(A, If[0 < u ≤ 1, 1 - u, 0], B) → A'
  24. Segment(A, A') → j_1
  25. Rotate(A, If[1 < u ≤ 2, ((u - 1) * α), α], A) → A'_1
  26. Rotate(A', If[1 < u ≤ 2, ((u - 1) * α), α], A) → A''
  27. Segment(A'_1, A'') → k_1
  28. Rotate(A'_1, If[2 < u ≤ 3, ((u - 2) * (π - α)), π - α], A) → Q
  29. Rotate(A'_1, If[2 < u ≤ 3, ((u - 2) * (π - α)), π - α], E) → R
  30. Rotate(A'', If[2 < u ≤ 3, ((u - 2) * (π - α)), π - α], E) → A'''_1
  31. Segment(R, A'''_1) → m_1
  32. Rotate(R, If[3 < u ≤ 4, ((u - 3) * α), α], C) → R'
  33. Rotate(A'''_1, If[3 < u ≤ 4, ((u - 3) * α), α], C) → S
  34. Segment(R', S) → l_1
  35. Polygon(A, B, C, E) → poly1, r_1, s_1, t_1, a_1
  36. Segment(E, D) → p_1
  37. Circle(A, (0.15 * f)) → d_1
  38. Segment(A, A'') → q_1
  39. Intersect(d_1, q_1) → T,
  40. Intersect(d_1, f) → U,
  41. CircleSector(A, U, T) → e_1
  42. Dilate(A, If[4 < u ≤ 5, 5 - u, 0], D) → A'_2
  43. Dilate(E, If[4 < u ≤ 5, 5 - u, 0], D) → E'
  44. Segment(A, A'_2) → b_1
  45. Segment(A'', E') → f_2
  46. Dilate(D, If[5 < u ≤ 6, 6 - u, 0], B) → D'
  47. Dilate(D, If[5 < u ≤ 6, 6 - u, 0], C) → D'_1
  48. Segment(D, D') → g_2
  49. Segment(D, D'_1) → h_2
  50. Dilate(D, If[6 < u ≤ 7, 7 - u, 0], G) → D'_2
  51. Segment(A, A'') → r_2
  52. Segment(A'', A) → b_2
  53. Segment(A, B) → f_3
  54. Segment(D, G) → g_3
  55. Segment(R, B) → h_3
  56. OrthogonalLine(D, l_1) → i_3
  57. OrthogonalLine(B, b_2) → j_3
  58. Intersect(b_2, j_3) → A_1
  59. Intersect(i_3, b_2) → B_1
  60. Segment(A_1, B_1) → l_3
  61. Point(l_3) → C_1
  62. OrthogonalLine(C_1, l_1) → p
  63. Intersect(p, l_1) → F
  64. Line(A, R) → r
  65. Line(B, A'') → t
  66. Intersect(p, r) → H
  67. Intersect(p, g_3) → I
  68. Intersect(t, p) → J
  69. Midpoint(H, C_1) → K
  70. Midpoint(J, F) → L
  71. Segment(C_1, K) → f_1
  72. Circle(I, f_1) → c_1
  73. Line(D, G) → g_1
  74. Intersect(c_1, g_1) → M, N
  75. Dilate(D, If[6 < u ≤ 7, 7 - u, 0], I) → D'_3
  76. Dilate(I, If[7 < u ≤ 8, 8 - u, 0], N) → I'
  77. Dilate(N, If[8 < u ≤ 9, 9 - u, 0], G) → N'
  78. Segment(D, D'_3) → h_1
  79. Segment(I, I') → i_1
  80. Segment(N, N') → n_1
  81. Dilate(C_1, If[9 < u ≤ 10, 10 - u, 0], K) → O
  82. Segment(C_1, O) → i_2
  83. Dilate(K, If[10 < u ≤ 11, 11 - u, 0], H) → K'
  84. Segment(K, K') → j_2
  85. Dilate(K, If[11 < u ≤ 12, 12 - u, 0], I) → K'_1
  86. Dilate(H, If[11 < u ≤ 12, 12 - u, 0], I) → H'
  87. Segment(H, H') → k_2
  88. Dilate(I, If[12 < u ≤ 13, 13 - u, 0], J) → I'_1
  89. Segment(I, I'_1) → l_2
  90. Dilate(J, If[13 < u ≤ 14, 14 - u, 0], L) → J'
  91. Segment(J, J') → m_2
  92. Dilate(L, If[14 < u ≤ 15, 15 - u, 0], F) → L'
  93. Segment(L, L') → n_2
  94. Angle(R, F, C_1) → γ
  95. Rotate(C_1, If[15 < u ≤ 16, ((-(u - 15)) * π), (-π)], K) → P
  96. Rotate(O, If[15 < u ≤ 16, ((-(u - 15)) * π), (-π)], K) → O'
  97. Segment(P, O') → p_2
  98. Rotate(J, If[16 < u ≤ 17, ((-(u - 16)) * π), (-π)], L) → J'_1
  99. Rotate(J', If[16 < u ≤ 17, ((-(u - 16)) * π), (-π)], L) → J''
  100. Segment(J'_1, J'') → q_2
  101. Dilate(J, If[17 < u ≤ 18, 18 - u, 0], G) → J'_2
  102. Vector(J, J'_2) → v
  103. Translate(J, v) → J'_3
  104. Translate(J', v) → J''_1
  105. Segment(J'_3, J''_1) → s_2
  106. Line(G, l_2) → t_2
  107. Point(t_2) → V
  108. Angle(V, G, D) → δ
  109. Rotate(J'_3, If[18 < u ≤ 19, ((u - 18) * δ), δ], G) → W
  110. Rotate(J''_1, If[18 < u ≤ 19, ((u - 18) * δ), δ], G) → Z
  111. Segment(W, Z) → a_2
  112. Line(H, N) → c_2
  113. Dilate(H, If[19 < u ≤ 20, 20 - u, 0], N) → H'_1
  114. Vector(H, H'_1) → w
  115. Translate(P, w) → P'
  116. Translate(O', w) → O''
  117. Segment(P', O'') → d_2
  118. Rotate(P', If[20 < u ≤ 21, ((-(u - 20)) * (π - δ)), δ - π], N) → P''
  119. Rotate(O'', If[20 < u ≤ 21, ((-(u - 20)) * (π - δ)), δ - π], N) → O'''
  120. Segment(P'', O''') → e_2