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Pythagoras' Theorem -- Area Dissection 1

Pythagoras' Theorem -- Area Dissection 1
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2017-12-11 15:53
TrianglePolygonTransformationGeometric Construction
In collections Mathematics Form 1
① This example demonstrates the Pythagorean theorem by dissecting and rearranging areas on the sides of a right triangle, visually showing that the sum of the squares on the legs equals the square on the hypotenuse. ② The core mathematical concept is the Pythagorean theorem and its area-based proof via dissection, involving plane figures such as triangles, squares, and semicircles. ③ Construction highlights: from the construction summary, the example uses GeoGebra commands including Semicircle, Midpoint, Point, Segment, Rotate, Ray, Intersect, Line, OrthogonalLine, Distance, Max, Min, Circle, Polygon, Vector, Translate, Angle, and SelectedIndex to build auxiliary lines, dissect regions, and dynamically compare areas. ④ Suitable for middle school geometry lessons or math outreach, helping students grasp the geometric proof of the Pythagorean theorem, and also useful as a reference for dynamic geometry worksheet design.

GeoGebra Commands & Parameters

  1. Semicircle(C, A) → c
  2. Midpoint(A, C) → C1
  3. Point(c) → B
  4. Segment(A, B) → a
  5. Segment(B, C) → b
  6. Segment(A, C) → d
  7. Rotate(A, 90°, B) → A2
  8. Rotate(B, -90°, A) → A1
  9. Segment(A, A1) → e
  10. Segment(A1, A2) → f
  11. Segment(A2, B) → g
  12. Rotate(B, 90°, C) → B2
  13. Rotate(C, -90°, B) → B1
  14. Segment(B, B1) → h
  15. Segment(B1, B2) → i
  16. Segment(B2, C) → j
  17. Rotate(A, -90°, C) → C2
  18. Rotate(C, 90°, A) → C3
  19. Segment(A, C3) → k
  20. Segment(C3, C2) → l
  21. Segment(C2, C) → m
  22. Rotate(A, -45°, B) → A'
  23. Ray(B, A') → n
  24. Intersect(d, n) → D
  25. Line(D, a) → p
  26. Line(D, b) → q
  27. Intersect(q, a) → A3
  28. Intersect(p, b) → B3
  29. Line(A3, k) → r
  30. Line(B3, m) → s
  31. OrthogonalLine(B3, s) → t
  32. Intersect(s, h) → B6
  33. Intersect(t, j) → B4
  34. Intersect(r, g) → A4
  35. OrthogonalLine(A4, r) → c_1
  36. OrthogonalLine(A3, r) → d_1
  37. Intersect(d_1, e) → A6
  38. Intersect(c_1, f) → A5
  39. OrthogonalLine(B6, s) → a_1
  40. Intersect(a_1, i) → B5
  41. Segment(A4, A5) → f_1
  42. Segment(A5, A6) → g_1
  43. Segment(B3, B4) → h_1
  44. Segment(B4, B5) → i_1
  45. Distance(A4, A5) → LenS1
  46. Distance(B4, B5) → LenS2
  47. Max(LenS1, LenS2) → BLen
  48. Min(LenS1, LenS2) → SLen
  49. Circle(A, LenS1) → q_1
  50. Intersect(q_1, k, 1) → C5
  51. Intersect(q_1, d, 1) → C8
  52. Line(C5, d) → l_1
  53. Line(C8, k) → m_1
  54. Intersect(m_1, l) → C7
  55. Intersect(l_1, m) → C6
  56. Intersect(l_1, m_1) → C4
  57. Line(C3, C4) → p_1
  58. Line(C, p_1) → r_1
  59. OrthogonalLine(C7, p_1) → n_1
  60. Line(C5, n_1) → s_1
  61. Segment(C8, C6) → t_1
  62. Intersect(p_1, t_1) → C10
  63. Intersect(r_1, t_1) → C9
  64. Intersect(p_1, n_1) → C11
  65. Intersect(p_1, s_1) → C12
  66. Segment(C3, C4) → a_2
  67. Segment(C4, C10) → b_2
  68. Segment(C, C9) → c_2
  69. Segment(C11, C7) → h_2
  70. Segment(C5, C12) → i_2
  71. Distance(A, A3) → LenT1
  72. Distance(A, A6) → LenT2
  73. Distance(B, B6) → LenT4
  74. Distance(B, B3) → LenT3
  75. Line(H, a) → s_2
  76. Line(H, b) → t_2
  77. Circle(H, LenT1) → c_3
  78. Circle(H, LenT2) → d_3
  79. Intersect(c_3, t_2, 2) → H2
  80. Intersect(d_3, s_2, 2) → H1
  81. Polygon(H, H1, H2) → poly4, h2, h_3, h1
  82. Line(I, a) → a_3
  83. Line(I, b) → b_3
  84. Circle(I, LenT1) → k_3
  85. Circle(I, LenT2) → p_3
  86. Intersect(p_3, b_3, 2) → I1
  87. Intersect(k_3, a_3, 1) → I2
  88. Polygon(I, I1, I2) → poly5, i2, i_3, i1
  89. Line(K, a) → m_3
  90. Line(K, b) → n_3
  91. Circle(K, LenT3) → s_3
  92. Circle(K, LenT4) → t_3
  93. Intersect(t_3, m_3, 1) → K2
  94. Intersect(s_3, n_3, 1) → K1
  95. Polygon(K, K2, K1) → poly7, k1, k_4, k2
  96. Line(L, a) → a_4
  97. Line(L, b) → b_4
  98. Circle(L, LenT3) → c_4
  99. Circle(L, LenT4) → d_4
  100. Intersect(c_4, b_4, 2) → L3
  101. Intersect(d_4, a_4, 2) → L2
  102. Line(M, a) → e_4
  103. Line(M, b) → f_4
  104. Circle(M, LenT3) → h_4
  105. Circle(M, LenT4) → p_4
  106. Intersect(h_4, e_4, 1) → M2
  107. Intersect(p_4, f_4, 2) → M1
  108. Line(N, a) → i_4
  109. Line(N, b) → n_4
  110. Circle(N, LenT3) → q_4
  111. Circle(N, LenT4) → r_4
  112. Intersect(q_4, i_4, 2) → N2
  113. Intersect(r_4, n_4, 1) → N1
  114. Polygon(N, N1, N2) → poly10, n2, n_5, n1
  115. Vector(I, C11) → u3
  116. Translate(I, v3) → I'
  117. Translate(I1, v3) → I1'
  118. Translate(I2, v3) → I2'
  119. Polygon(I', I1', I2') → poly5', i2', i', i1'
  120. Vector(H, C9) → u4
  121. Vector(K, C12) → u6
  122. Translate(K, v6) → K'
  123. Translate(K2, v6) → K2'
  124. Translate(K1, v6) → K1'
  125. Polygon(K', K2', K1') → poly7', k1', k', k2'
  126. Vector(N, C9) → u9
  127. Translate(N, v9) → N'
  128. Translate(N1, v9) → N1'
  129. Translate(N2, v9) → N2'
  130. Polygon(N', N1', N2') → poly10', n2', n', n1'
  131. Translate(H, v5) → H'
  132. Translate(H1, v5) → H1'
  133. Translate(H2, v5) → H2'
  134. Polygon(H', H1', H2') → poly4', h2', h', h1'
  135. OrthogonalLine(E, b) → b_1
  136. OrthogonalLine(E, a) → e_1
  137. Circle(E, LenT1) → k_1
  138. Circle(E, Distance[A, B]) → d_2
  139. Intersect(k_1, e_1, 1) → E1
  140. Intersect(d_2, b_1, 2) → E2
  141. Line(E2, b) → j_1
  142. Circle(E2, LenT2) → e_2
  143. Intersect(e_2, j_1, 1) → E3
  144. Line(E3, m) → f_2
  145. Line(E1, d) → g_2
  146. Intersect(f_2, g_2) → E4
  147. Polygon(E, E1, E4, E3, E2) → poly1, e_3, e1, g_3, f_3, e2
  148. Line(F, b) → j_2
  149. Line(F, a) → k_2
  150. Circle(F, Distance[B, C]) → p_2
  151. Intersect(p_2, k_2, 1) → F1
  152. Circle(F, LenT4) → q_2
  153. Intersect(q_2, j_2, 2) → F3
  154. Line(F1, b) → l_2
  155. Circle(F1, LenT3) → r_2
  156. Intersect(r_2, l_2, 2) → F4
  157. Line(F4, d) → m_2
  158. Line(F3, k) → n_2
  159. Intersect(m_2, n_2) → F5
  160. Polygon(F, F1, F4, F5, F3) → poly2, f_5, f1, f4, f5, f3
  161. Vector(E, C10) → u1
  162. Translate(E, v1) → E'
  163. Translate(E1, v1) → E1'
  164. Translate(E4, v1) → E4'
  165. Translate(E3, v1) → E3'
  166. Translate(E2, v1) → E2'
  167. Polygon(E', E1', E4', E3', E2') → poly1', e', e1', e4', e3', e2'
  168. Vector(F, C10) → u2
  169. Translate(F, v2) → F'
  170. Translate(F1, v2) → F1'
  171. Translate(F4, v2) → F4'
  172. Translate(F5, v2) → F5'
  173. Translate(F3, v2) → F3'
  174. Polygon(F', F1', F4', F5', F3') → poly2', f', f1', f4', f5', f3'
  175. Circle(B, S90) → q_3
  176. Intersect(q_3, a, 1) → G
  177. Intersect(q_3, b, 1) → J
  178. Rotate(B, -90°, J) → B'
  179. Translate(E, w1) → E'_1
  180. Translate(E1, w1) → E1'_1
  181. Translate(E4, w1) → E4'_1
  182. Translate(E3, w1) → E3'_1
  183. Translate(E2, w1) → E2'_1
  184. Polygon(E'_1, E1'_1, E4'_1, E3'_1, E2'_1) → poly1'_1, e'_1, e1'_1, e4'_1, e3'_1, e2'_1
  185. Translate(F, w2) → F'_1
  186. Translate(F1, w2) → F1'_1
  187. Translate(F4, w2) → F4'_1
  188. Translate(F5, w2) → F5'_1
  189. Translate(F3, w2) → F3'_1
  190. Polygon(F'_1, F1'_1, F4'_1, F5'_1, F3'_1) → poly2'_1, f'_1, f1'_1, f4'_1, f5'_1, f3'_1
  191. Translate(I, w3) → I'_1
  192. Translate(I1, w3) → I1'_1
  193. Translate(I2, w3) → I2'_1
  194. Polygon(I'_1, I1'_1, I2'_1) → poly5'_1, i2'_1, i'_1, i1'_1
  195. Translate(H, w4) → H'_1
  196. Translate(H1, w4) → H1'_1
  197. Translate(H2, w4) → H2'_1
  198. Polygon(H'_1, H1'_1, H2'_1) → poly4'_1, h2'_1, h'_1, h1'_1
  199. Translate(K, w6) → K'_1
  200. Translate(K2, w6) → K2'_1
  201. Translate(K1, w6) → K1'_1
  202. Polygon(K'_1, K2'_1, K1'_1) → poly7'_1, k1'_1, k'_1, k2'_1
  203. Translate(N, w9) → N'_1
  204. Translate(N1, w9) → N1'_1
  205. Translate(N2, w9) → N2'_1
  206. Polygon(N'_1, N1'_1, N2'_1) → poly10'_1, n2'_1, n'_1, n1'_1
  207. Angle(C, B, A) → α
  208. SelectedIndex(Bahasa) → nn