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Area of Triangles: Dynamic Construction and Comparison

Area of Triangles: Dynamic Construction and Comparison
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2014-01-02 06:33
TrianglePolygonAngleTransformationGeometric Construction
In collections AREA
This GeoGebra example dynamically demonstrates the area of triangles, focusing on right triangles and general triangles, and explores how transformations such as rotation and reflection affect area. The core mathematical concepts include the triangle area formula (base × height / 2), congruence, symmetry, and the use of auxiliary lines (altitudes, medians) to decompose or rearrange shapes. The construction uses commands like Line, OrthogonalLine, Point, Intersect, Segment, Midpoint, Semicircle, Rotate, Angle, Polygon, and Mirror to build multiple triangles and polygons, measure their areas, and label angles. It is suitable for middle school to high school students to visually grasp area derivation, triangle congruence criteria, and geometric transformations, and can serve as a classroom demonstration or self-guided exploration tool.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. OrthogonalLine(A, a) → b
  3. Point(b) → C
  4. Line(C, a) → c
  5. OrthogonalLine(B, a) → d
  6. Intersect(c, d) → D
  7. Segment(A, B) → e
  8. Point(e) → E
  9. Segment(E, C) → f
  10. Segment(E, D) → g
  11. Midpoint(f) → F
  12. Midpoint(g) → G
  13. Semicircle(E, D) → h
  14. OrthogonalLine(E, e) → i
  15. Intersect(c, i) → H
  16. Semicircle(B, H) → k
  17. Semicircle(D, E) → p_1
  18. Semicircle(H, A) → q
  19. Semicircle(C, E) → r
  20. Semicircle(E, C) → s
  21. Rotate(E, -α, G) → E'
  22. Angle(E', G, E) → β
  23. Rotate(B, -α, G) → B'
  24. Angle(B', G, B) → γ
  25. Rotate(D, -α, G) → D'
  26. Angle(D', G, D) → δ
  27. Polygon(E', B', D') → poly1, d', e', b'
  28. Rotate(A, α, F) → A'
  29. Angle(A, F, A') → ε
  30. Rotate(E, α, F) → E'_1
  31. Angle(E, F, E'_1) → ζ
  32. Rotate(C, α, F) → C'
  33. Angle(C, F, C') → η
  34. Polygon(A', C', E'_1) → poly2, e'_1, a', c'
  35. Midpoint(e) → I
  36. Midpoint(A, C) → J
  37. Midpoint(C, D) → K
  38. Midpoint(B, D) → L
  39. Polygon(J, I, L, K) → poly3, j_1, i_1, l, k_1
  40. Midpoint(j_1) → M
  41. Midpoint(i_1) → N
  42. Midpoint(l) → O
  43. Midpoint(k_1) → P
  44. Rotate(J, α, P) → J'
  45. Angle(J, P, J') → θ
  46. Rotate(C, α, P) → C'_1
  47. Angle(C, P, C'_1) → ι
  48. Rotate(K, α, P) → K'
  49. Angle(K, P, K') → κ
  50. Polygon(J', C'_1, K') → poly4, k', j', c'_1
  51. Rotate(K, -α, O) → K'_1
  52. Angle(K'_1, O, K) → λ
  53. Rotate(L, -α, O) → L'
  54. Angle(L', O, L) → μ
  55. Rotate(D, -α, O) → D'_1
  56. Angle(D'_1, O, D) → ν
  57. Polygon(K'_1, D'_1, L') → poly5, l', k'_1, d'_1
  58. Rotate(A, -α, M) → A'_1
  59. Angle(A'_1, M, A) → ξ
  60. Rotate(I, -α, M) → I'
  61. Angle(I', M, I) → ο
  62. Rotate(J, -α, M) → J'_1
  63. Angle(J'_1, M, J) → ρ
  64. Polygon(J'_1, A'_1, I') → poly6, i', j'_1, a'_1
  65. Rotate(B, α, N) → B'_1
  66. Angle(B, N, B'_1) → σ
  67. Rotate(L, α, N) → L'_1
  68. Angle(L, N, L'_1) → τ
  69. Rotate(I, α, N) → I'_1
  70. Angle(I, N, I'_1) → υ
  71. Polygon(B'_1, L'_1, I'_1) → poly7, i'_1, b'_1, l'_1
  72. Segment(A', C') → n_2
  73. Polygon(C, E, D) → poly8, d_1, c_1, e_1
  74. Midpoint(A, C) → Q
  75. Midpoint(D, C) → R
  76. Midpoint(A, B) → S
  77. Midpoint(B, D) → T
  78. Segment(Q, T) → t
  79. Segment(S, R) → a_1
  80. Intersect(t, a_1) → U
  81. Point(e) → V
  82. Mirror(V, U) → V'
  83. Polygon(C, Q, V, B, T, V') → poly9, c_2, q_1, v, b_1, t_1, v'
  84. Rotate(A, α, Q) → A'_2
  85. Angle(A, Q, A'_2) → ϕ
  86. Rotate(V, α, Q) → V'_1
  87. Angle(V, Q, V'_1) → χ
  88. Polygon(Q, V'_1, A'_2) → poly10, a'_2, q_2, v'_1
  89. Rotate(V', α, T) → V''
  90. Angle(V', T, V'') → ψ
  91. Rotate(D, α, T) → D'_2
  92. Angle(D, T, D'_2) → ω
  93. Polygon(V'', D'_2, T) → poly11, t_2, v'', d'_2
  94. Segment(C, D) → f_1
  95. Segment(C, V') → g_1
  96. Segment(V, B) → h_1
  97. Segment(Q, C) → l_1
  98. Midpoint(A, C) → W
  99. Midpoint(B, D) → Z
  100. Segment(V, B) → m_1
  101. Point(m_1) → A_1
  102. Polygon(W, C, D, Z, A_1, V) → poly12, w, c_3, d_2, z, u_2, v_1
  103. Rotate(A_1, -α, Z) → B_1
  104. Angle(B_1, Z, A_1) → α_1
  105. Rotate(B, -α, Z) → B'_2
  106. Angle(B'_2, Z, B) → β_1
  107. Polygon(B_1, Z, B'_2) → poly13, b'_2, b_2, z_1
  108. Segment(E, C) → n_1
  109. Segment(C, D) → r_1
  110. Segment(E, D) → s_1
  111. Midpoint(s_1) → C_1
  112. Rotate(E, α, C_1) → E'_2
  113. Angle(E, C_1, E'_2) → γ_1
  114. Rotate(C, α, C_1) → C'_2
  115. Angle(C, C_1, C'_2) → δ_1
  116. Rotate(D, α, C_1) → D'_3
  117. Angle(D, C_1, D'_3) → ε_1
  118. Polygon(E'_2, C'_2, D'_3) → poly14, d'_3, e'_2, c'_2
  119. Midpoint(A, C) → D_1
  120. Midpoint(B, D) → E_1
  121. Mirror(V, D_1) → V'_2
  122. Mirror(A_1, E_1) → F_1
  123. Polygon(V, V'_2, F_1, A_1) → poly15, v_2, v'_2, f_2, a_2
  124. Midpoint(A_1, F_1) → G_1
  125. Rotate(A_1, α, G_1) → H_1
  126. Angle(A_1, G_1, H_1) → ζ_1
  127. Rotate(V, α, G_1) → V'_3
  128. Angle(V, G_1, V'_3) → η_1
  129. Rotate(V'_2, α, G_1) → I_1
  130. Angle(V'_2, G_1, I_1) → θ_1
  131. Rotate(F_1, α, G_1) → J_1
  132. Angle(F_1, G_1, J_1) → ι_1
  133. Polygon(V'_3, I_1, J_1, H_1) → poly16, v'_3, i_2, j_2, h_2
  134. Polygon(E'_1, A', C') → poly17, c'_3, e'_3, a'_3
  135. Polygon(A, C, D, B) → poly18, a_3, c_4, d_3, b_3
  136. Polygon(C, E, D) → poly19, d_4, c_5, e_2
  137. Polygon(E', B', D') → poly20, d'_4, e'_4, b'_3
  138. Polygon(A'_1, J'_1, I') → poly21, i'_2, a'_4, j'_2
  139. Polygon(B'_1, I'_1, L'_1) → poly22, l'_2, b'_4, i'_3
  140. Polygon(J', K', C'_1) → poly23, c'_4, j'_3, k'_2
  141. Polygon(K'_1, D'_1, L') → poly24, l'_3, k'_3, d'_5
  142. Polygon(A, C, D, B) → poly25, a_4, c_6, d_5, b_4
  143. Polygon(I, J, K, L) → poly26, i_3, j_3, k_2, l_2
  144. Polygon(V'_1, A'_2, Q) → poly27, q_3, v'_4, a'_5
  145. Polygon(D'_2, T, V'') → poly28, v''_1, d'_6, t_3
  146. Polygon(V, V'_1, V', V'') → poly29, v_3, v'_5, v'_6, v''_2
  147. Polygon(A, C, D, B) → poly30, a_5, c_7, d_6, b_5
  148. Point(b_5) → K_1
  149. Polygon(V'_1, A'_2, Q) → poly31, q_4, v'_7, a'_6
  150. Polygon(B_1, Z, B'_2) → poly32, b'_5, b_6, z_2
  151. Polygon(V, V'_1, B_1, A_1) → poly33, v_4, v'_8, b_7, a_6
  152. Polygon(A, C, D, B) → poly34, a_7, c_8, d_7, b_8
  153. Polygon(E, C, D) → poly35, d_8, e_3, c_9
  154. Polygon(D'_3, E'_2, C'_2) → poly36, c'_5, d'_7, e'_5
  155. Polygon(C'_2, E, C, D) → poly37, c'_6, e_4, c_{10}, d_9
  156. Polygon(V, V'_2, F_1, A_1) → poly38, v_5, v'_9, f_3, a_8
  157. Polygon(J_1, H_1, V'_3, I_1) → poly39, j_4, h_3, v'_{10}, i_4
  158. Polygon(V, V'_2, V'_3, I_1) → poly40, v_6, v'_{11}, v'_{12}, i_5

AI Drawing Prompt Reference

Draw a right triangle and a general triangle to demonstrate the calculation of their areas. Show how the area changes when these triangles are rotated and reflected. Include auxiliary lines like altitudes and medians to decompose or rearrange the shapes. Measure and label the angles, comparing the areas and shapes of the transformed figures.
The prompt above was compiled from the original drawing commands and is for reference only.