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Exploring Scaled Area Calculations

Exploring Scaled Area Calculations
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2024-02-19 20:47
TriangleCirclePolygonGeometric Construction
This example demonstrates how scaling factors affect the area of similar geometric figures. The key mathematical concept is the relationship between area and the square of length ratios. Construction involves using GeoGebra commands like Dilate, Scale, and Polygon to create and analyze shapes. Suitable for students in grades 6-8 to understand area scaling and solve related problems.

GeoGebra Commands & Parameters

  1. Shuffle(Sequence[1.5, 4, 0.5]) → l3, {2, 1.5, 3.5, 4, 2.5, 3}
  2. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.4685917812681518, 0.2012505003582646, 0.6861007067706905, 0.7399771036234561, 0.5591575070748879, 0.967307700310622, 0.3344672936141103, 0.8943741054129235, 0.12411893941661649, 0.7283439071931139, 0.7134459892295615, 0.7864478715997133, 0.9203324373289585, 0.1595312876003502, 0.43681860632545055, 0.6994790703703223, 0.35134653522982107, 0.5499721830362213, 0.9103485558805878, 0.5510033577989492, 0.6048676903862722, 0.5921390587985159, 0.7575876936910675, 0.05148036983408587, 0.793971967693963, 0.5540596716554305, 0.891436020368879, 0.3535190718641712, 0.44072196148771203, 0.3514492629118613, 0.82436585781155, 0.9429187897321395, 0.3386509145000225, 0.7112784119856784, 0.8153699003646058, 0.4781662692300122, 0.3705666897088904, 0.3422407906584341, 0.9440256570602621, 0.6450974429490559, 0.8703053441786598, 0.06124564842870794, 0.2415903308836257, 0.6281885189314631, 0.7414298021768069, 0.9384110406963908, 0.06574970893159936, 0.06938401378860264, 0.2924214477922616, 0.8760696682306596, 0.03954061461394963, 0.3995009827943604, 0.2166533170231289, 0.16002467049332267, 0.4960506552064089, 0.25606597099571166, 0.7263603252086491, 0.6237043253953077, 0.4087702786910239, 0.5449482089518884, 0.012938305789364737, 0.3879643307453672, 0.2729316245682983, 0.25134493768203625, 0.6312054005845202, 0.1510736469292123, 0.1958145734453649, 0.09855642334518795, 0.6527504098920882, 0.1848942204047093, 0.7838347169781993, 0.9705559381185194, 0.22316963074146184, 0.32442712428701687, 0.9612493849845878, 0.516616291802054, 0.43985969953039983, 0.5087474800879143, 0.03952670479549401, 0.09319762860231386, 0.321086163891386, 0.20188696041989657, 0.687819593022338, 0.5199320847192918, 0.16328797375481313, 0.4024483863169672, 0.6735698216285801, 0.9028593577547502, 0.7320058148132909, 0.16064779415856115, 0.8780396171578498, 0.21907307679728905, 0.9624372384472262, 0.06108004719424442, 0.6045265019798202, 0.8126309790558239, 0.5784247115191367, 0.7787902977270894, 0.5471642789076123, 0.6819632982542471, 0.35367240929864446, 0.7152089086388086, 0.8755461713260696, 0.8040814306216163, 0.24616569341690553, 0.39916919124722483, 0.3649287300689395, 0.20064543308825267, 0.5761322789533799, 0.9165767201527276, 0.09348280955430642, 0.03539195339759896, 0.9194368823933075, 0.49033690037968636, 0.8469557001102006, 0.6415548849256614, 0.10204144873087889, 0.9398709520735944, 0.5771914685303815, 0.5117299849024585, 0.8870906011400892, 0.44696360011614766, 0.5242284959962898, 0.8811398281521996, 0.7842973452307263, 0.5317847484703585, 0.09215238558358689, 0.42844941529352387, 0.9216066723606151, 0.3497146717144147, 0.2665282653067051, 0.11616331649777815, 0.0016052508902960616, 0.23607713472388178, 0.7160710728390852, 0.585203545307498, 0.5898226462143195, 0.7520464854401782, 0.38323387491799965, 0.4255234659254453, 0.7689350926207821, 0.930844704094749, 0.29760263324892866, 0.006556263197856094, 0.271935439576263, 0.6888815287317953, 0.14238311526199024, 0.18792723078986093, 0.7585733085230425, 0.4217152294049767, 0.5278491479464251, 0.7121776990857893, 0.15279685333256798, 0.29962359329745786, 0.6863930882344745, 0.8972542548084345, 0.6245432196377763, 0.04217482982026742, 0.5517781128598126, 0.24721010250437642, 0.5670270898613022, 0.8454743136759684, 0.6903388944499601, 0.664946396947328, 0.0006673668768865537, 0.7997389544575821, 0.7342106880731758, 0.3308899433902417, 0.40696933438164323, 0.09851157161646086, 0.09618863281472143, 0.4162588900641588, 0.2800452917230724, 0.5341786431549556, 0.852528114050382, 0.8290122699693009, 0.4500090428760303, 0.13749111044081402, 0.7738932011473344, 0.5247092266066127, 0.8612201397076797, 0.8315125842909481, 0.005744564012955949, 0.9733229918674869, 0.8892812731829061, 0.28283196963306345, 0.13772941976816616, 0.5265595595590833, 0.5438075726299297, 0.014976216084488231, 0.8270951775196388, 0.2984253948078233, 0.41928939954568645, 0.6207469436741405, 0.7237949889151893, 0.944668602295427, 0.16386022838213452, 0.2616786936815658, 0.04478740439312818, 0.9242865035922172}
  3. Corner(5) → AppSize
  4. Polygon({(-1.5, -0.5), (10.5, -0.5), (10.5, 10.5), (-1.5, 10.5)}) → graphWin
  5. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  6. If(2 + floor((currentRandomSeeds(4) * 7)) ≟ base, If[base ≟ 2, 3, base - 1], 2 + floor((currentRandomSeeds(4) * 7))) → hei
  7. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  8. Dilate(B, scaleFactor, A) → B'
  9. Translate(Polygon[A, B', 4], (3, 0)) → poly2
  10. Translate(A, (0, -1)) → A'
  11. Translate(B, (0, -1)) → B'_{1}
  12. Segment(A', B'_{1}) → f'
  13. Translate(A, (0, -1)) → A'_{1}
  14. Translate(B', (0, -1)) → B''
  15. Segment(Translate[A'_{1}, (3, 0)], Translate[B'', (3, 0)]) → j'
  16. Text((Length[f] * "\text{ in.}"), Midpoint[f'], true, true, 0, -1) → labelOrigSq
  17. Text((Length[j'] * "\;in."), Midpoint[j'], true, true, 0, -1) → labelDilatedSq
  18. Translate(hGuideline, Vector[(Vector[Corner[1], Corner[4]] / 500 * 16)]) → hGuideline'
  19. Zip((("$\textsf{" * tt) * " }$"), tt, l1) → l2
  20. If(currentRandomSeeds(2) ≤ 0.5, 2, 3) → Case
  21. Polygon(A, B, (x(B), y(C1)), C1) → q1, n, p, q, r
  22. Translate(I, Vector[(Vector[E, F] * scaleFactor)]) → I'
  23. Translate(I', Vector[(Vector[F, G] * scaleFactor)]) → I''
  24. Element({4, 3, 3, 2.5}, Case) → maxSL
  25. Polygon(I, I', I'', J) → q2, i_1, i', i'', j
  26. Polygon({A, B, C1}) → t1
  27. Polygon({I, I', I''}) → t2
  28. Translate(A, Vector[(-0.5, 0)]) → H'
  29. Translate(A, Vector[(-0.5, 0)]) → E'
  30. Segment(E', (x(E'), y(C1))) → k
  31. Translate(I', Vector[(0.5, 0)]) → I''_{1}
  32. Translate(I'', Vector[(0.5, 0)]) → I'''
  33. Segment(I''_{1}, I''') → i''_{1}
  34. Text((k * "\text{ in.}"), Midpoint[k], true, true, -1, 0) → text1
  35. Text((i''_{1} * "\textsf{ in.}"), Midpoint[i''_{1}], true, true, 1, 0) → text2
  36. Circle(K, L) → c
  37. Circle(M, (Distance[K, L] * scaleFactor)) → d
  38. Segment(K, L) → l
  39. Text(legendOF, Centroid[Element[{poly1, q1, t1, Polygon[K + (-1, -1.5), K + (1, -1.5), 4]}, Case]], true, true, 0, 0) → textAreaOF
  40. Text(legendDF, Centroid[Element[{poly2, q2, t2, Polygon[M + (-1, -1.5), M + (1, -1.5), 4]}, Case]], true, true, 0, 0) → textAreaSF
  41. Segment(M, M + (Radius[d], 0)) → m
  42. Text((l * "\textsf{ in.}"), Midpoint[l], true, true, 0, 1) → text3
  43. Text((m * "\textsf{ in.}"), Midpoint[m], true, true, 0, 1) → text4
  44. Polygon(D1, D1 + (boxWidth, 0), D1 + (boxWidth, boxHeight), D1 + (0, boxHeight)) → box1, s, t, a, c_2
  45. Polygon(T2, T2 + (1.22, 0), T2 + (1.22, 0.95), T2 + (0, 0.95)) → blank2, h_1, j_1, k_1, l_1
  46. Translate(blank2, (1.9, 0)) → blank3
  47. Polygon(D3, D3 + (boxWidth, 0), D3 + (boxWidth, boxHeight), D3 + (0, boxHeight)) → box3, d3, c_1, d_1, e_1
  48. Polygon(D2, D2 + (boxWidth, 0), D2 + (boxWidth, boxHeight), D2 + (0, boxHeight)) → box2, d2, f_2, g_2, h_2
  49. Vertex(blank3, 1) → T3
  50. Polygon(D4, D4 + ((2 * boxWidth), 0), D4 + ((2 * boxWidth), boxHeight), D4 + (0, boxHeight)) → box4, i_2, j_2, k_2, d_2
  51. Polygon(D5, D5 + ((2 * boxWidth), 0), D5 + ((2 * boxWidth), boxHeight), D5 + (0, boxHeight)) → box5, s_2, t_2, a_2, b_2
  52. Text("A", B, true, true, 1, 1) → text5
  53. Polygon(T1, T1 + (1.22, 0), T1 + (1.22, 0.95), T1 + (0, 0.95)) → blank1, b, e, f_1, g_1
  54. Translate(vGuideline, Vector[(Vector[Corner[1], Corner[2]] / 800 * 24)]) → vGuideline'
  55. Polygon(Home2_{1}, Home2_{1} + (1.22, 0), Home2_{1} + (1.22, 0.95), Home2_{1} + (0, 0.95)) → q3, home2, m_1, n_1, p_1
  56. Polygon(Home2, Home2 + (1.22, 0), Home2 + (1.22, 0.95), Home2 + (0, 0.95)) → q4, home3, q_1, r_1, s_1
  57. Polygon(Home3, Home3 + (1.22, 0), Home3 + (1.22, 0.95), Home3 + (0, 0.95)) → q5, home4, t_1, a_1, b_1
  58. Translate(blank1, (2.5, 0)) → blank1'
  59. Vertex(blank1', 1) → Home4
  60. Polygon(T4, T4 + (boxWidth, 0), T4 + (boxWidth, boxHeight), T4 + (0, boxHeight)) → q6, t4, l_2, m_2, n_2
  61. Polygon(T5, T5 + ((2 * boxWidth), 0), T5 + ((2 * boxWidth), boxHeight), T5 + (0, boxHeight)) → q7, t5, p_2, q_2, r_2
  62. Text(scaleFactor, Centroid[box1] + (0, 0.15), true, true, 0, 0) → textBox1
  63. Text(text1, Centroid[box3] + (0, 0.15), true, true, 0, 0) → textBox3
  64. Text(labelOrigSq, Centroid[box2] + (0, 0.15), true, true, 0, 0) → textBox2
  65. Defined(IndexOf[true, Zip[IsInRegion[PP, blank1], PP, {Vertex[box1]}]]) → box1InReg
  66. Defined(IndexOf[true, Zip[IsInRegion[PP, blank2], PP, {Vertex[box2]}]]) → box2InReg
  67. Defined(IndexOf[true, Zip[IsInRegion[PP, blank3], PP, {Vertex[box3]}]]) → box3InReg
  68. Polygon({T4, T4 + ((2 * boxWidth), 0), T4 + ((2 * boxWidth), boxHeight), T4 + (0, boxHeight)}) → blank4
  69. Polygon({T5, T5 + ((2 * boxWidth), 0), T5 + ((2 * boxWidth), boxHeight), T5 + (0, boxHeight)}) → blank5
  70. Defined(IndexOf[true, Zip[IsInRegion[PP, blank4], PP, {Vertex[box4]}]]) → box4InReg
  71. Defined(IndexOf[true, Zip[IsInRegion[PP, blank5], PP, {Vertex[box5]}]]) → box5InReg
  72. Text(scaleFactor^(2), Centroid[box4] + (0, 0.1), true, true, 0, 0) → textBox4
  73. Polygon(D1 + (0.1, 0), D1 - (0.2, 0) + (boxWidth, 0), D1 - (0.2, 0) + (boxWidth, boxHeight), D1 + (0.1, 0) + (0, boxHeight)) → box1Vis, e_2, f_3, g_3, h_3
  74. IndexOf(true, {box1InReg ∧ box1InMove, box2InReg ∧ box2InMove, box3InReg ∧ box3InMove, box4InReg ∧ box4InMove, box5InReg ∧ box5InMove}) → indexHL
  75. Element({blank1, blank2, blank3, blank4, blank5}, indexHL) → q8
  76. onboardingTextOffset(Corner[onboardingText_{1}, 1], Corner[onboardingText_{1}, 3], onboardingCase_{1}, Corner[1], Corner[3], Corner[5]) → OnboardingTextOffset_{1}
  77. expandQuad(Vertex[box1, 1], Vertex[box1, 2], Vertex[box1, 3], Vertex[box1, 4], (7 * pulsing), Cor1, Cor3, (800, 500)) → pulsingObject1
  78. expandQuad(Vertex[box2, 1], Vertex[box2, 2], Vertex[box2, 3], Vertex[box2, 4], (7 * pulsing), Cor1, Cor3, (800, 500)) → pulsingObject2
  79. expandQuad(Vertex[box3, 1], Vertex[box3, 2], Vertex[box3, 3], Vertex[box3, 4], (7 * pulsing), Cor1, Cor3, (800, 500)) → pulsingObject3
  80. expandQuad(Vertex[box4, 1], Vertex[box4, 2], Vertex[box4, 3], Vertex[box4, 4], (7 * pulsing), Cor1, Cor3, (800, 500)) → pulsingObject4
  81. expandQuad(Vertex[box5, 1], Vertex[box5, 2], Vertex[box5, 3], Vertex[box5, 4], (7 * pulsing), Cor1, Cor3, (800, 500)) → pulsingObject5
  82. Text((((base * hei) * If[Case ≟ 3, 1 / 2, 1]) * "\text{ in^2}"), Centroid[box5] + (0, 0.15), true, true, 0, 0) → textBox5

AI Drawing Prompt Reference

Draw a geometric illustration demonstrating how scaling factors affect the area of similar figures. Start with a rectangle as the background, then randomly generate a quadrilateral inside it and calculate its area. Next, dilate the quadrilateral using the same scale factor and calculate the area of the enlarged quadrilateral. Label the original and dilated areas next to the shapes, using different colors for distinction. Finally, include a circle to show the relationship between its radius and area, emphasizing the connection between scale factors and the square of the area.
The prompt above was compiled from the original drawing commands and is for reference only.