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Pythagorean Theorem (Special Case) Dynamic Demonstration

Pythagorean Theorem (Special Case) Dynamic Demonstration
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2023-06-05 07:19
TrianglePolygonTransformationGeometric ConstructionSlider Animation
① This GeoGebra example dynamically demonstrates a special case of the Pythagorean theorem, where the two legs of a right triangle are equal, visually verifying that the square on the hypotenuse equals the sum of the squares on the other two sides. ② The core mathematical concept is the Pythagorean theorem, involving right triangles, areas of squares, rotations and translations, and the construction of auxiliary lines to prove area equality. ③ Construction highlights: Based on the construction summary, commands such as Point, Segment, Polygon, Rotate, Ray, Circle, Intersect, Vector, Translate, Midpoint, OrthogonalLine, LineBisector, and Line are used. Multiple points (A, B, C, etc.), segments (a, b, c), polygons (q1, q2), and rotated images (A', B', etc.) are created to form squares on each side and demonstrate the area relationship through rotation. ④ Suitable for middle school geometry teaching, helping students grasp the geometric meaning of the Pythagorean theorem, and useful for classroom demonstrations or self-exploration by adjusting the triangle shape to observe area changes.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Point(yAxis) → B
  3. Point(xAxis) → C
  4. Segment(B, C) → g
  5. Segment(C, A) → h
  6. Point(xAxis) → G
  7. Polygon(H, G, B, A) → q1, h_1, g_1, b, a
  8. Rotate(H, 0.6435, A) → H'_{1}
  9. Rotate(G, 0.6435, A) → G'_{1}
  10. Rotate(B, 0.6435, A) → B'_{2}
  11. Rotate(A, 0.6435, A) → A'_{2}
  12. Rotate(H, c, A) → H'
  13. Rotate(G, c, A) → G'
  14. Rotate(B, c, A) → B'
  15. Rotate(A, c, A) → A'
  16. Polygon(H', G', B', A') → q1', h', g', b', a'
  17. Ray(G'_{1}, B'_{2}) → i
  18. Polygon(H'_{1}, G'_{1}, B'_{2}, A) → q2, h'_{1}, g'_{1}, b'_{2}, a_1
  19. Circle(G'_{1}, c - 0.64) → d
  20. Intersect(d, i) → D,
  21. Vector(G'_{1}, D) → u
  22. Translate(H'_{1}, u) → E
  23. Translate(G'_{1}, u) → F
  24. Translate(B'_{2}, u) → I
  25. Translate(A, u) → A'_{1}
  26. Polygon(E, F, I, A'_{1}) → q2', e, f_1, i_1, a'_{1}
  27. Segment(H', G') → f
  28. Segment(G', B') → j
  29. Segment(B', A) → k
  30. Segment(A, H') → m
  31. Midpoint(H', A) → J
  32. Midpoint(G', B') → K
  33. Midpoint(H', J) → L
  34. Midpoint(G', K) → M
  35. Midpoint(K, B') → N
  36. Midpoint(J, A) → O
  37. Midpoint(H', G') → Q
  38. Midpoint(H', Q) → P
  39. Midpoint(Q, G') → R
  40. Midpoint(A, B') → S
  41. Midpoint(A, S) → T
  42. Midpoint(S, B') → U
  43. Segment(L, M) → l
  44. Segment(J, K) → n
  45. Segment(O, N) → p
  46. Segment(P, T) → q
  47. Segment(Q, S) → r
  48. Segment(R, U) → s
  49. Midpoint(A'_{1}, E) → V
  50. Midpoint(I, D) → W
  51. Midpoint(V, E) → Z
  52. Midpoint(A'_{1}, V) → A_1
  53. Midpoint(I, W) → B_1
  54. Midpoint(W, D) → C_1
  55. Segment(Z, C_1) → t
  56. Segment(V, W) → j_1
  57. Segment(A_1, B_1) → k_1
  58. Midpoint(E, D) → D_1
  59. Midpoint(D_1, D) → E_1
  60. Midpoint(E, D_1) → F_1
  61. Midpoint(A'_{1}, I) → G_1
  62. Midpoint(A'_{1}, G_1) → H_1
  63. Midpoint(G_1, I) → I_1
  64. Segment(F_1, H_1) → l_1
  65. Segment(D_1, G_1) → m_1
  66. Segment(E_1, I_1) → n_1
  67. Point(yAxis) → J_1
  68. Polygon(B, M_1, L_1, C) → q3, b_1, p_1, q_1, c_1
  69. Intersect(i, h) → O_1
  70. OrthogonalLine(N_1, i) → r_1
  71. Intersect(r_1, i) → P_1
  72. Circle(P_1, 1) → d_1
  73. Intersect(d_1, r_1, 2) → Q_1
  74. LineBisector(N_1, Q_1) → s_1
  75. LineBisector(C, B) → t_1
  76. Intersect(s_1, t_1) → R_1
  77. Rotate(B, c - 4.64, R_1) → B'_{1}
  78. Rotate(M_1, c - 4.64, R_1) → S_1
  79. Rotate(L_1, c - 4.64, R_1) → T_1
  80. Rotate(C, c - 4.64, R_1) → C'
  81. Polygon(B'_{1}, S_1, T_1, C') → q3', b'_{1}, e_1, f_2, c'
  82. Polygon(M_1, W_1, Z_1, L_1) → q4, h_2, w_1, z_1, i_2
  83. Intersect(d_1, i, 1) → A_2
  84. Midpoint(O_1, A_2) → B_2
  85. Midpoint(M_1, L_1) → C_2
  86. Segment(A_2, L_1) → j_2
  87. Rotate(M_1, c - 6.85, M_1) → E_2
  88. Rotate(W_1, c - 6.85, M_1) → F_2
  89. Rotate(Z_1, c - 6.85, M_1) → G_2
  90. Rotate(L_1, c - 6.85, M_1) → H_2
  91. Polygon(E_2, F_2, G_2, H_2) → q4', e_2, k_2, l_2, m_2
  92. OrthogonalLine(M_1, u) → n_2
  93. OrthogonalLine(C, n_2) → p_2
  94. Intersect(n_2, p_2) → I_2
  95. Polygon(M_1, I_2, C, O_1) → q5, q_2, r_2, c_2, o_1
  96. Circle(I_2, c - 7.49) → d_2
  97. Intersect(d_2, p_2) → J_2, K_2
  98. Vector(I_2, J_2) → v
  99. Translate(M_1, v) → L_2
  100. Translate(I_2, v) → M_2
  101. Translate(C, v) → C'_{1}
  102. Translate(O_1, v) → N_2
  103. Polygon(L_2, M_2, C'_{1}, N_2) → q5', s_2, t_2, c'_{1}, a_2
  104. Polygon(A, O_1, P_1, N_1) → q6, a_3, b_2, f_3, g_3
  105. Midpoint(N_1, A) → O_2
  106. Midpoint(N_1, O_2) → P_2
  107. Midpoint(O_2, A) → Q_2
  108. Midpoint(P_1, O_1) → R_2
  109. Midpoint(O_1, R_2) → S_2
  110. Midpoint(N_1, P_1) → T_2
  111. Midpoint(N_1, T_2) → U_2
  112. Midpoint(T_2, A) → V_2
  113. Midpoint(O_1, A) → W_2
  114. Midpoint(A, W_2) → Z_2
  115. Segment(Q_2, S_2) → h_3
  116. Segment(O_2, R_2) → i_3
  117. Segment(P_2, A_2) → j_3
  118. Segment(U_2, Z_2) → k_3
  119. Segment(W_2, T_2) → l_3
  120. Midpoint(W_2, O_1) → A_3
  121. Segment(Q_1, A_3) → m_3
  122. Intersect(d_1, i) → B_3, C_3
  123. OrthogonalLine(C_3, i) → n_3
  124. OrthogonalLine(N_1, r_1) → p_3
  125. Intersect(n_3, p_3) → D_3
  126. Circle(D_3, 1) → c_3
  127. Intersect(c_3, n_3, 1) → E_3
  128. Circle(E_3, 1) → d_3
  129. Intersect(d_3, n_3, 1) → F_3
  130. Circle(F_3, 1) → e_3
  131. Intersect(e_3, n_3, 1) → G_3
  132. Circle(N_1, 1) → q_3
  133. Intersect(d_3, q_3, 1) → H_3
  134. Circle(H_3, 1) → r_3
  135. Intersect(e_3, r_3, 1) → I_3
  136. Polygon(N_1, Q_1, G_3, D_3) → q7, s_3, t_3, b_3, f_4
  137. Segment(E_3, H_3) → g_4
  138. Segment(F_3, I_3) → h_4
  139. Segment(C', B'_{1}) → i_4
  140. Segment(B'_{1}, S_1) → j_4
  141. Segment(S_1, T_1) → k_4
  142. Segment(T_1, C') → l_4
  143. Circle(C', 1) → c_4
  144. Circle(T_1, 1) → d_4
  145. Intersect(c_4, i_4, 1) → J_3
  146. Intersect(d_4, k_4, 1) → K_3
  147. Circle(S_1, 1) → e_4
  148. Intersect(e_4, k_4, 1) → L_3
  149. Circle(B'_{1}, 1) → p_4
  150. Intersect(p_4, i_4) → M_3,
  151. Segment(M_3, L_3) → m_4
  152. Segment(J_3, K_3) → n_4
  153. Circle(C'_{1}, 1) → q_4
  154. Intersect(q_4, t_2, 1) → N_3
  155. Circle(N_2, 1) → r_4
  156. Intersect(a_2, b_2) → O_3
  157. Intersect(r_4, a_2) → P_3,
  158. Segment(P_3, N_3) → s_4
  159. Circle(J_2, 1) → t_4
  160. Circle(L_2, 1) → c_5
  161. Intersect(t_4, t_2, 1) → Q_3
  162. Intersect(a_2, b_2) → R_3
  163. Intersect(c_5, a_2) → S_3,
  164. Segment(S_3, Q_3) → a_4
  165. Polygon(W_1, J_1, T_3, V_3) → q8, b_4, f_5, g_5, v_3
  166. Circle(C_3, 1) → d_5
  167. Intersect(d_5, n_3) → Z_3, A_4
  168. Segment(V_3, L_1) → h_5
  169. Polygon(G_4, L_1, K_1, T_3, V_3, W_3) → poly1, m_5, n_5, p_5, q_5, r_5, w_3
  170. Segment(W_3, P_1) → s_5
  171. Circle(W_3, c - 12.73) → t_5
  172. Intersect(t_5, s_5, 1) → B_4
  173. Vector(W_3, B_4) → w
  174. Translate(G_4, w) → C_4
  175. Translate(L_1, w) → D_4
  176. Translate(K_1, w) → E_4
  177. Translate(T_3, w) → F_4
  178. Translate(V_3, w) → H_4
  179. Translate(W_3, w) → I_4
  180. Polygon(C_4, D_4, E_4, F_4, H_4, I_4) → poly1', i_5, j_5, k_5, l_5, a_5, b_5
  181. Circle(P_1, sqrt(2)) → g_2
  182. Line(W_3, K_1) → e_5
  183. Vector(W_3, P_1) → u_1
  184. Translate(G_4, u_1) → U_1
  185. Translate(L_1, u_1) → V_1
  186. Translate(K_1, u_1) → J_4
  187. Translate(T_3, u_1) → K_4
  188. Translate(V_3, u_1) → L_4
  189. Translate(W_3, u_1) → M_4
  190. Polygon(U_1, V_1, J_4, K_4, L_4, M_4) → poly1'_{1}, f_6, v_1, g_6, h_6, i_6, j_6
  191. Rotate(U_1, c - 19.05, P_1) → N_4
  192. Rotate(V_1, c - 19.05, P_1) → O_4
  193. Rotate(J_4, c - 19.05, P_1) → P_4
  194. Rotate(K_4, c - 19.05, P_1) → Q_4
  195. Rotate(L_4, c - 19.05, P_1) → R_4
  196. Rotate(M_4, c - 19.05, P_1) → S_4
  197. Polygon(N_4, O_4, P_4, Q_4, R_4, S_4) → poly2, k_6, o_4, l_6, m_6, n_6, p_6
  198. Midpoint(O_4, P_4) → T_4
  199. Segment(P_1, T_4) → q_6
  200. Midpoint(Q_4, P_4) → U_4
  201. Segment(P_1, U_4) → r_6
  202. Midpoint(D_4, E_4) → V_4
  203. Midpoint(F_4, E_4) → W_4
  204. Segment(B_4, W_4) → s_6
  205. Segment(B_4, V_4) → t_6
  206. Segment(M_1, H_2) → a_6
  207. Segment(H_2, G_2) → b_6
  208. Segment(G_2, F_2) → c_6
  209. Segment(F_2, M_1) → d_6
  210. Circle(H_2, 1) → e_6
  211. Intersect(e_6, a_6, 1) → Z_4
  212. Circle(G_2, 1) → c_7
  213. Intersect(c_7, c_6, 1) → A_5
  214. Circle(M_1, 1) → d_7
  215. Intersect(d_7, a_6, 1) → B_5
  216. Circle(F_2, 1) → e_7
  217. Intersect(e_7, c_6, 1) → C_5
  218. Segment(B_5, C_5) → f_7
  219. Segment(Z_4, A_5) → g_7
  220. Segment(W_3, U_3) → h_7
  221. Segment(W_3, Z_1) → i_7
  222. Segment(M_1, G_4) → j_7
  223. Midpoint(R_2, O_1) → D_5
  224. OrthogonalLine(C, h) → k_7
  225. OrthogonalLine(B_3, k_7) → l_7
  226. Intersect(k_7, l_7) → E_5
  227. OrthogonalLine(R_2, k_7) → m_7
  228. OrthogonalLine(D_5, k_7) → n_7
  229. Intersect(k_7, n_7) → F_5
  230. Intersect(k_7, m_7) → G_5
  231. Polygon(O_1, C, E_5, B_3) → q9, p_7, c_8, q_7, r_7
  232. Segment(D_5, F_5) → s_7
  233. Segment(R_2, G_5) → t_7
  234. Circle(V_3, c - 10.49) → d_8
  235. Intersect(d_8, h_5, 1) → D_2
  236. Vector(V_3, D_2) → u_2
  237. Translate(W_1, u_2) → H_5
  238. Translate(J_1, u_2) → I_5
  239. Translate(T_3, u_2) → J_5
  240. Translate(V_3, u_2) → K_5
  241. Polygon(H_5, I_5, J_5, K_5) → q8', a_7, b_7, f_8, g_8
  242. Polygon(V_3, T_3, K_1, Z_1) → q10, h_8, i_8, j_8, k_8
  243. Segment(H, G) → l_8
  244. Segment(G, B) → m_8
  245. Segment(B, A) → n_8
  246. Segment(H, A) → p_8
  247. Midpoint(H, A) → L_5
  248. Midpoint(A, B) → M_5
  249. Midpoint(B, G) → N_5
  250. Midpoint(H, G) → O_5
  251. Midpoint(H, O_5) → P_5
  252. Midpoint(O_5, G) → Q_5
  253. Midpoint(A, M_5) → R_5
  254. Midpoint(M_5, B) → S_5
  255. Midpoint(G, N_5) → T_5
  256. Midpoint(N_5, B) → U_5
  257. Midpoint(H, L_5) → V_5
  258. Midpoint(L_5, A) → W_5
  259. Segment(V_5, T_5) → q_8
  260. Segment(L_5, N_5) → r_8
  261. Segment(W_5, U_5) → s_8
  262. Segment(Q_5, S_5) → t_8
  263. Segment(O_5, M_5) → a_8
  264. Segment(P_5, R_5) → b_8
  265. Segment(M_1, L_1) → e_8
  266. Point(g) → Z_5
  267. Point(g) → A_6
  268. Segment(Z_5, V_3) → f_9
  269. Segment(A_6, W_3) → g_9
  270. Segment(A, B) → h_9
  271. Segment(B, C) → i_9
  272. Segment(C, A) → j_9
  273. Polygon(B_6, E_6, D_6, C_6) → q11, k_9, l_9, m_9, n_9

AI Drawing Prompt Reference

Draw a right triangle ABC with two equal legs. Construct squares q1 and q2 with A, B, and C as vertices. Rotate and translate squares q1 and q2 to align their corresponding sides, forming new squares q1' and q2'. Use auxiliary lines and midpoints to demonstrate that the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides.
The prompt above was compiled from the original drawing commands and is for reference only.