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Triangle Area Demonstration by GreenMaths

Triangle Area Demonstration by GreenMaths
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2013-11-28 02:00
TriangleCoordinate GeometryGeometric Construction
In collections Area Investigations Area - Rectangles, Triangles and Parallelograms
This example illustrates the connection between the area of a triangle and the area of a rectangle using an acute triangle. Key math concepts include triangle area formula, rectangle area formula, height, median, rotation, etc. Constructed with GeoGebra commands like Point, Circle, Segment, OrthogonalLine, Intersect, Polygon, Midpoint, Rotate. Suitable for middle school students and above to learn the relationship between areas of geometric shapes, or for math teaching demonstrations.

GeoGebra Commands & Parameters

  1. Point(Circle[A, Height]) → C
  2. Circle(A, C) → e
  3. Point(Circle[A, Base]) → B
  4. Segment(A, B) → b
  5. OrthogonalLine(B, b) → g
  6. OrthogonalLine(A, b) → f
  7. Intersect(e, f, 1) → D
  8. OrthogonalLine(D, f) → h
  9. Intersect(g, h) → E
  10. Polygon(A, D, E, B) → P, a_1, d_1, e_1, b_1
  11. Point(d_1) → S_2
  12. OrthogonalLine(S_2, b_1) → a_5
  13. Segment(A, C) → d
  14. Midpoint(A, D) → F
  15. Midpoint(A, B) → G
  16. Point(Circle[A, 1]) → H
  17. Segment(A, H) → i
  18. Point(Circle[A, 2]) → I
  19. Segment(A, I) → j
  20. Point(Circle[A, 3]) → J
  21. Segment(A, J) → k
  22. Point(Circle[A, 4]) → K
  23. Segment(A, K) → l
  24. Point(Circle[A, 5]) → L
  25. Segment(A, L) → m
  26. Point(Circle[A, 6]) → M
  27. Segment(A, M) → n
  28. Point(Circle[A, 7]) → N
  29. Segment(A, N) → p
  30. Point(Circle[A, 8]) → O
  31. Segment(A, O) → q
  32. Point(Circle[A, 9]) → Q
  33. Segment(A, Q) → r
  34. Circle(A, H) → s
  35. Circle(A, I) → t
  36. Circle(A, J) → c_1
  37. Circle(A, K) → f_1
  38. Circle(A, L) → g_1
  39. Circle(A, M) → h_1
  40. Circle(A, N) → k_1
  41. Circle(A, O) → p_1
  42. Circle(A, Q) → q_1
  43. Intersect(s, a_1, 1) → R
  44. Intersect(t, a_1, 1) → S
  45. Intersect(c_1, a_1, 1) → T
  46. Intersect(f_1, a_1, 1) → U
  47. Intersect(g_1, a_1, 1) → V
  48. Intersect(h_1, a_1, 1) → W
  49. Intersect(k_1, a_1, 1) → Y
  50. Intersect(p_1, a_1, 1) → Z
  51. Intersect(q_1, a_1, 1) → A_1
  52. Intersect(q_1, b_1, 1) → B_1
  53. Intersect(p_1, b_1, 1) → C_1
  54. Intersect(k_1, b_1, 1) → D_1
  55. Intersect(h_1, b_1, 1) → E_1
  56. Intersect(g_1, b_1, 1) → F_1
  57. Intersect(f_1, b_1, 1) → G_1
  58. Intersect(c_1, b_1, 1) → H_1
  59. Intersect(t, b_1) → I_1,
  60. Intersect(s, b_1, 1) → J_1
  61. OrthogonalLine(R, a_1) → i_1
  62. OrthogonalLine(S, a_1) → j_1
  63. OrthogonalLine(T, a_1) → l_1
  64. OrthogonalLine(U, a_1) → m_1
  65. OrthogonalLine(V, a_1) → n_1
  66. OrthogonalLine(W, a_1) → r_1
  67. OrthogonalLine(Y, a_1) → s_1
  68. OrthogonalLine(Z, a_1) → t_1
  69. OrthogonalLine(A_1, a_1) → a_2
  70. OrthogonalLine(J_1, b_1) → b_2
  71. OrthogonalLine(I_1, b_1) → c_2
  72. OrthogonalLine(H_1, b_1) → d_2
  73. OrthogonalLine(G_1, b_1) → e_2
  74. OrthogonalLine(F_1, b_1) → f_2
  75. OrthogonalLine(E_1, b_1) → g_2
  76. OrthogonalLine(D_1, b_1) → h_2
  77. OrthogonalLine(C_1, b_1) → i_2
  78. OrthogonalLine(B_1, b_1) → j_2
  79. Intersect(b_2, d_1) → K_1
  80. Intersect(c_2, d_1) → L_1
  81. Intersect(d_2, d_1) → M_1
  82. Intersect(e_2, d_1) → N_1
  83. Intersect(f_2, d_1) → O_1
  84. Intersect(g_2, d_1) → P_1
  85. Intersect(h_2, d_1) → Q_1
  86. Intersect(i_2, d_1) → R_1
  87. Intersect(j_2, d_1) → S_1
  88. Intersect(a_2, e_1) → T_1
  89. Intersect(t_1, e_1) → U_1
  90. Intersect(s_1, e_1) → V_1
  91. Intersect(r_1, e_1) → W_1
  92. Intersect(n_1, e_1) → Y_1
  93. Intersect(m_1, e_1) → Z_1
  94. Intersect(l_1, e_1) → A_2
  95. Intersect(j_1, e_1) → B_2
  96. Intersect(i_1, e_1) → C_2
  97. Segment(J_1, K_1) → k_2
  98. Segment(I_1, L_1) → l_2
  99. Segment(H_1, M_1) → m_2
  100. Segment(G_1, N_1) → n_2
  101. Segment(F_1, O_1) → p_2
  102. Segment(E_1, P_1) → q_2
  103. Segment(D_1, Q_1) → r_2
  104. Segment(C_1, R_1) → s_2
  105. Segment(B_1, S_1) → t_2
  106. Segment(R, C_2) → a_3
  107. Segment(B_2, S) → b_3
  108. Segment(T, A_2) → c_3
  109. Segment(Z_1, U) → d_3
  110. Segment(V, Y_1) → e_3
  111. Segment(W_1, W) → f_3
  112. Segment(Y, V_1) → g_3
  113. Segment(U_1, Z) → h_3
  114. Segment(A_1, T_1) → i_3
  115. Intersect(q_1, a_1) → D_2, D_5
  116. Intersect(p_1, a_1) → E_2, E_5
  117. Intersect(k_1, a_1) → F_2, F_4
  118. Intersect(h_1, a_1) → G_2, G_4
  119. Intersect(g_1, a_1) → H_2, H_4
  120. Intersect(f_1, a_1) → I_2, I_4
  121. Intersect(c_1, a_1) → J_2, J_4
  122. Intersect(t, a_1) → K_2, K_3
  123. Intersect(c_1, b_1) → L_2, L_3
  124. Intersect(f_1, b_1) → M_2, M_3
  125. Intersect(g_1, b_1) → N_2, N_3
  126. Intersect(h_1, b_1) → O_2, O_3
  127. Intersect(k_1, b_1) → P_2, P_3
  128. Intersect(q_1, b_1) → Q_2, Q_3
  129. Intersect(p_1, b_1) → R_2, R_3
  130. OrthogonalLine(D_2, a_1) → a
  131. OrthogonalLine(E_2, a_1) → c
  132. OrthogonalLine(F_2, a_1) → j_3
  133. OrthogonalLine(G_2, a_1) → k_3
  134. OrthogonalLine(H_2, a_1) → l_3
  135. OrthogonalLine(I_2, a_1) → m_3
  136. OrthogonalLine(J_2, a_1) → n_3
  137. OrthogonalLine(K_2, a_1) → p_3
  138. OrthogonalLine(L_2, b_1) → q_3
  139. OrthogonalLine(M_2, b_1) → r_3
  140. OrthogonalLine(N_2, b_1) → s_3
  141. OrthogonalLine(O_2, b_1) → t_3
  142. OrthogonalLine(P_2, b_1) → a_4
  143. OrthogonalLine(R_2, b_1) → b_4
  144. OrthogonalLine(Q_2, b_1) → c_4
  145. Intersect(q_3, d_1) → U_2
  146. Intersect(r_3, d_1) → V_2
  147. Intersect(s_3, d_1) → W_2
  148. Intersect(t_3, d_1) → Y_2
  149. Intersect(a_4, d_1) → Z_2
  150. Intersect(b_4, d_1) → A_3
  151. Intersect(c_4, d_1) → B_3
  152. Intersect(e_1, a) → C_3
  153. Intersect(e_1, c) → D_3
  154. Intersect(e_1, j_3) → E_3
  155. Intersect(e_1, k_3) → F_3
  156. Intersect(e_1, l_3) → G_3
  157. Intersect(e_1, m_3) → H_3
  158. Intersect(e_1, n_3) → I_3
  159. Intersect(p_3, e_1) → J_3
  160. Segment(U_2, L_2) → d_4
  161. Segment(M_2, V_2) → e_4
  162. Segment(W_2, N_2) → f_4
  163. Segment(O_2, Y_2) → g_4
  164. Segment(Z_2, P_2) → h_4
  165. Segment(R_2, A_3) → i_4
  166. Segment(B_3, Q_2) → j_4
  167. Segment(J_3, K_2) → k_4
  168. Segment(J_2, I_3) → l_4
  169. Segment(H_3, I_2) → m_4
  170. Segment(H_2, G_3) → n_4
  171. Segment(F_3, G_2) → p_4
  172. Segment(F_2, E_3) → q_4
  173. Segment(D_3, E_2) → r_4
  174. Segment(D_2, C_3) → s_4
  175. Intersect(a_1, b_1) → T_2
  176. Intersect(e_1, b_1) → S_3
  177. Intersect(b_1, a_5) → T_3
  178. Polygon(T_2, S_2, T_3) → poly1, t_4, t_5, s_5
  179. Polygon(S_2, S_3, T_3) → poly2, t_6, s_6, s_7
  180. Midpoint(t_6) → V_3
  181. Midpoint(t_4) → W_3
  182. Rotate(S_2, α, V_3) → Z_3
  183. Rotate(S_3, α, V_3) → A_4
  184. Rotate(T_3, α, V_3) → B_4
  185. Polygon(Z_3, A_4, B_4) → poly2', b_5, z_3, a_6
  186. Rotate(T_2, -α, W_3) → C_4
  187. Rotate(S_2, -α, W_3) → D_4
  188. Rotate(T_3, -α, W_3) → E_4
  189. Polygon(C_4, D_4, E_4) → poly1', e_5, c_5, d_5

AI Drawing Prompt Reference

Draw an acute triangle and demonstrate the relationship between its area and a related rectangle. Start by drawing a triangle with one vertex and base. Then find the perpendicular bisector of the base. Next, draw circles centered at the vertex with different radii, intersecting the bisector at multiple points. Connect these points to the triangle's vertices and base to form several small triangles and rectangles. Finally, use rotation and color differentiation to clearly show the area relationship between the triangle and rectangles.
The prompt above was compiled from the original drawing commands and is for reference only.