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Effects of Scaling on Areas of Similar Figures

Effects of Scaling on Areas of Similar Figures
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2018-05-20 09:30
Geometric Construction
In collections Perimeter & Area
This example demonstrates the changes in area of similar figures when one figure is scaled up by a scale factor 'k'. Key math concepts include similarity transformation and area ratio relationships. The construction uses the Dilate command to create scaled figures and Polygon objects to visualize area changes. Suitable for high school students to understand properties of similar figures and ratio relationships, serving as an aid in geometric construction and transformation teaching.

GeoGebra Commands & Parameters

  1. Rotate(B, (a * π / 180), A) → B'
  2. Angle(B, A, B') → α
  3. Segment(A, B) → f
  4. Ray(A, B') → g
  5. Point(g) → C
  6. Segment(A, C) → h
  7. Segment(C, B) → i
  8. Line(A, C) → j
  9. Line(C, B) → k
  10. Line(A, B) → l
  11. Circle(C, A) → c
  12. Intersect(c, j, 2) → D
  13. Circle(D, C) → d
  14. Intersect(d, j, 2) → E
  15. Line(D, i) → m
  16. Line(E, i) → n
  17. Intersect(l, m) → F
  18. Intersect(l, n) → G
  19. Line(C, l) → p
  20. Line(D, l) → q
  21. Intersect(m, p) → H
  22. Intersect(n, q) → I
  23. Line(B, H) → r
  24. Line(F, r) → s
  25. Polygon(A, B, C) → t1, c_1, a_1, b
  26. Rotate(A, If[0 < e ≤ 1, (e * π), π], C) → A'
  27. Segment(C, A') → t
  28. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π), (-π)], B) → A'_1
  29. Segment(B, A'_1) → f_1
  30. Dilate(C, If[2 < e ≤ 3, 3 - e, 0], D) → C'
  31. Vector(C, C') → u
  32. Translate(C, u) → C'_1
  33. Translate(B, u) → B'_1
  34. Segment(C'_1, B'_1) → i'
  35. Dilate(B, If[3 < e ≤ 4, 4 - e, 0], F) → C'_2
  36. Vector(B, C'_2) → v
  37. Translate(C, v) → C'_3
  38. Translate(B, v) → B'_2
  39. Segment(C'_3, B'_2) → i'_1
  40. Dilate(C, If[4 < e ≤ 5, 5 - e, 0], H) → C'_4
  41. Segment(C, C'_4) → g_1
  42. Dilate(B, If[4 < e ≤ 5, 5 - e, 0], H) → B'_3
  43. Segment(B, B'_3) → h_1
  44. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'_5
  45. Vector(C, C'_5) → w
  46. Translate(A, w) → A'_2
  47. Translate(B, w) → B'_4
  48. Translate(C, w) → C'_6
  49. Polygon(A'_2, B'_4, C'_6) → t1', c'_6, a'_2, b'_4
  50. Dilate(D, If[7.75 < e ≤ 8.75, 8.75 - e, 0], H) → D'
  51. Vector(D, D') → u_1
  52. Translate(A'_2, u_1) → J
  53. Translate(B'_4, u_1) → K
  54. Translate(C'_6, u_1) → L
  55. Polygon(J, K, L) → t1'', l_1, j_1, k_1
  56. Midpoint(h_1) → M
  57. Rotate(J, If[9.5 < e ≤ 10.5, ((e - 9.5) * π), π], M) → J'
  58. Rotate(K, If[9.5 < e ≤ 10.5, ((e - 9.5) * π), π], M) → K'
  59. Rotate(L, If[9.5 < e ≤ 10.5, ((e - 9.5) * π), π], M) → L'
  60. Polygon(J', K', L') → t1''', l', j', k'
  61. Rotate(C, If[13 < e ≤ 14, ((e - 13) * π), π], D) → C'_7
  62. Segment(D, C'_7) → i_1
  63. Midpoint(I, G) → N
  64. Rotate(B, If[13 < e ≤ 14, ((-(e - 13)) * π), (-π)], F) → B'_5
  65. Segment(F, B'_5) → m_1
  66. Dilate(D, If[14 < e ≤ 15, 15 - e, 0], E) → D'_1
  67. Vector(D, D'_1) → v_1
  68. Translate(C'_1, v_1) → O
  69. Translate(B'_1, v_1) → P
  70. Segment(O, P) → i''
  71. Translate(C'_3, v_1) → Q
  72. Translate(B'_2, v_1) → R
  73. Segment(Q, R) → n_1
  74. Dilate(F, If[15 < e ≤ 16, 16 - e, 0], G) → F'
  75. Vector(F, F') → w_1
  76. Translate(C'_3, w_1) → S
  77. Translate(B'_2, w_1) → T
  78. Segment(S, T) → p_1
  79. Dilate(H, If[16 < e ≤ 17, 17 - e, 0], N) → H'
  80. Dilate(H, If[16 < e ≤ 17, 17 - e, 0], I) → H'_1
  81. Dilate(D, If[16 < e ≤ 17, 17 - e, 0], I) → D'_2
  82. Dilate(F, If[16 < e ≤ 17, 17 - e, 0], N) → F'_1
  83. Segment(F, F'_1) → q_1
  84. Segment(H, H') → r_1
  85. Segment(H, H'_1) → s_1
  86. Segment(D, D'_2) → t_1
  87. Dilate(D, If[17.75 < e ≤ 18.75, 18.75 - e, 0], E) → D'_3
  88. Vector(D, D'_3) → b_1
  89. Dilate(F, If[17.75 < e ≤ 18.75, 18.75 - e, 0], G) → F'_2
  90. Vector(F, F'_2) → d_1
  91. Translate(A'_2, b_1) → U
  92. Translate(B'_4, b_1) → V
  93. Translate(C'_6, b_1) → W
  94. Polygon(U, V, W) → t1''_1, w_2, u_2, v_2
  95. Translate(J, d_1) → J'_1
  96. Translate(K, d_1) → K'_1
  97. Translate(L, d_1) → L'_1
  98. Polygon(J'_1, K'_1, L'_1) → t1'''_1, l'_1, j'_1, k'_1
  99. Dilate(N, If[19.5 < e ≤ 20.5, 20.5 - e, 0], I) → N'
  100. Vector(N, N') → e_1
  101. Translate(J'_1, e_1) → Z
  102. Translate(K'_1, e_1) → A_1
  103. Translate(L'_1, e_1) → B_1
  104. Polygon(Z, A_1, B_1) → t2, b_2, z_1, a_2
  105. Midpoint(r_1) → C_1
  106. Rotate(Z, If[21.25 < e ≤ 22.25, ((e - 21.25) * π), π], C_1) → Z'
  107. Rotate(A_1, If[21.25 < e ≤ 22.25, ((e - 21.25) * π), π], C_1) → D_1
  108. Rotate(B_1, If[21.25 < e ≤ 22.25, ((e - 21.25) * π), π], C_1) → E_1
  109. Polygon(Z', D_1, E_1) → t2', e_2, z', d_2
  110. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], D) → C'_8
  111. Vector(C, C'_8) → c_2
  112. Translate(J', c_2) → J''
  113. Translate(K', c_2) → K''
  114. Translate(L', c_2) → L''
  115. Polygon(J'', K'', L'') → t3, l'', j'', k''