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Identifying Right Triangles Using the Pythagorean Theorem

Identifying Right Triangles Using the Pythagorean Theorem
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2022-11-09 23:16
TriangleGeometric Construction
This example demonstrates how to use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle. Key mathematical concepts include the Pythagorean Theorem and its converse. Construction involves using GeoGebra commands like RandomUniform, Polygon, and Angle to create triangles and verify side relationships. Suitable for students in grades 6-8 for geometry learning and hands-on practice.

GeoGebra Commands & Parameters

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.8277290589633475, 0.560564223819123, 0.46710213174405857, 0.2904621745861907, 0.23743610299557338, 0.9431959434407235, 0.4446986026001729, 0.46480875235586894, 0.1775844984350674, 0.5684223835312537, 0.1270465511473419, 0.5787084265515502, 0.9853842256048307, 0.5548093691175524, 0.2909738220953605, 0.5873439320190732, 0.20112255535622156, 0.5610143311077863, 0.3712522922149391, 0.6963293941431112, 0.1511163919655492, 0.6494434582541048, 0.9371940815263081, 0.35592599744677955, 0.22318190971188748, 0.6682326378643612, 0.7527161610111212, 0.35902407163567135, 0.11944145478416895, 0.7275227025615276, 0.5203335380695489, 0.5785247439231074, 0.7490519173858466, 0.5461844267089139, 0.03402318954709579, 0.885408671764353, 0.32213969199831305, 0.8137721995225117, 0.11632109272632342, 0.8478713409886073, 0.07586554156799644, 0.5095455721207632, 0.35137192654966737, 0.4644334006627005, 0.47412000791528897, 0.9358960539801607, 0.41794606037481896, 0.7199015552522717, 0.5038623353243321, 0.7574520008820556, 0.44467090270350573, 0.06245959936213339, 0.9435680800227041, 0.5395076192642156, 0.7838934738072175, 0.7830113964530546, 0.7626386820188181, 0.46023249460478255, 0.1706984742903164, 0.5141763034544332, 0.7549527706456522, 0.7203661946480402, 0.22965141192073418, 0.12605480817435433, 0.025002297434413342, 0.7017880036227687, 0.15381810531474369, 0.2990364268746459, 0.9012581853215692, 0.522474961891458, 0.17853248038117167, 0.8205131315679983, 0.5185484141887358, 0.6699218806679175, 0.476241535450767, 0.7823919680815933, 0.09930814386160847, 0.5780949030561228, 0.12855156201340226, 0.5797659561151114, 0.9425745268983611, 0.4944260512761868, 0.5773317980094524, 0.20800799465281086, 0.0695506598757992, 0.883229091416547, 0.4084021201348739, 0.5458931816693763, 0.8654512588326628, 0.6202273301885664, 0.798938218090601, 0.5461351069901744, 0.04916932012411274, 0.5396325180897323, 0.7032731352061782, 0.9519378099210977, 0.8850662063829613, 0.7203991293508427, 0.3212559820855126, 0.06881999700760968, 0.09488907833387578, 0.6119016470525086, 0.10874906865589826, 0.22505926739935425, 0.41245827045536787, 0.41134275940135134, 0.34120184073565174, 0.21163403115546633, 0.30934318519593507, 0.605698706271955, 0.7585346908433072, 0.4735070215611151, 0.13625920571982006, 0.3199432973357357, 0.8027988098291978, 0.0884991712338863, 0.20236892407758508, 0.11066535755857049, 0.48924739567445297, 0.7678438421440472, 0.6330601402341609, 0.3200047016827708, 0.026561526629238694, 0.7837911992301185, 0.6660989700536333, 0.5690802439184087, 0.8430187429462281, 0.46434888976814975, 0.7163197742475345, 0.9127685946746635, 0.8923746319185818, 0.21339989155547823, 0.40021584263959603, 0.3988154331936571, 0.5261478583100341, 0.41602985856967173, 0.35393173778718423, 0.11382184268645197, 0.7478565364820436, 0.18392571798493648, 0.6512364394781209, 0.706646101870305, 0.12954845405786558, 0.06417419995487987, 0.36206810349984186, 0.23619714052174173, 0.8717004772184824, 0.006692895076910595, 0.6971939074564576, 0.4654683672012606, 0.6042320047334492, 0.5308688534111489, 0.24041643846282187, 0.5969717529476303, 0.2889851848230591, 0.770758288239417, 0.41658406716439256, 0.866479825429727, 0.012569025173133436, 0.32041837131752315, 0.7114273693077846, 0.9321685674235328, 0.362947459163956, 0.07659613236557719, 0.7879434775207511, 0.19487766738082704, 0.2535708448337441, 0.18509106284624122, 0.943690096004807, 0.9522648105807785, 0.3052084948491782, 0.8396380135721171, 0.3439265113754492, 0.4251288480694608, 0.0679337743795753, 0.2981227414783022, 0.487052673752153, 0.5281123832436724, 0.6239467030482266, 0.6003478828098298, 0.7012485067869776, 0.44936701194266837, 0.37608827229190334, 0.8444079003785897, 0.3546047712341493, 0.29457520342926646, 0.09700852538875443, 0.2950117577858102, 0.8753621066196811, 0.5999711741057907, 0.7299948673277691, 0.5568079812904525, 0.9041395255385412, 0.7266876965184035, 0.8144501113862676, 0.40892197985540596, 0.926234065066884, 0.8048803354493896, 0.5569292306852489, 0.8435063915455362}
  2. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  3. If(countQuestion ≟ 0, 16, Element[Sequence[A^(2), A, 4, 15], 1 + floor((currentRandomSeeds(5) * 12))]) → randomSqLengthA
  4. If(countQuestion ≟ 0, 25, Element[Sequence[A^(2), A, 4, 15], 1 + floor((currentRandomSeeds(6) * 12))]) → randomSqLengthB
  5. If(countQuestion ≟ 0, 1, 1 + floor((currentRandomSeeds(2) * 2))) → randomRight
  6. If(randomSqLengthA ≟ randomSqLengthB, 1, randomRight) → rightTriangle
  7. If(randomSolveType ≟ 1, randomSqLengthA + randomSqLengthB, abs(randomSqLengthA - randomSqLengthB)) → correctThirdSquared
  8. Element({-4, -3, -2, -1, 1, 2, 3, 4}, 1 + floor((currentRandomSeeds(1) * 8))) → randomAdd
  9. Sort({randomSqLengthA, randomSqLengthB, If[rightTriangle ≟ 1, correctThirdSquared, incorrectThirdSquared]}) → tripleSquaredList
  10. PixelsToPoint((xMin, yMin), (xMax, yMax), (800, 500), (465, 155)) → PAnchor
  11. Polygon({(0, 0), (boxWidth, 0), (boxWidth, boxHeight), (0, boxHeight)}) → boxTemplate
  12. centerBox(boxWidth, boxHeight, PointPyth2) → Pyth2
  13. centerBox(boxWidth, boxHeight, PointPyth3) → Pyth3
  14. centerBox(boxWidth, boxHeight, PointPyth1) → Pyth1
  15. Zip(simpRadForm[a], a, tripleSquaredList) → tripleListText
  16. Zip(currentLocations(A), A, {2, 3, 1}) → studentSelections
  17. Zip(tripleSquaredList(IndexOf[A, studentSelections]), A, {1, 2, 3}) → studentValues
  18. Defined(studentValues(3)) → filledRegion3
  19. Defined(studentValues(2)) → filledRegion2
  20. Defined(studentValues(1)) → filledRegion1
  21. PixelsToPoint((xMin, yMin), (xMax, yMax), (800, 500), (280, 300)) → PtC
  22. Defined(IndexOf[3, currentLocations]) → t3Filled
  23. Defined(IndexOf[2, currentLocations]) → t2Filled
  24. Defined(IndexOf[1, currentLocations]) → t1Filled
  25. Sequence(If[¬(filledBool(n)), pythCenters(n), (-100, -100)], n, 1, 3) → pythCentersUse
  26. If(countQuestion ≟ 0, pi / 3, Element[{pi / 3, (2 * pi / 3), (4 * pi / 3), (5 * pi / 3)}, 1 + floor((currentRandomSeeds(3) * 4))]) → randomRotation
  27. Rotate(PtC + (tripleList(2), 0), cornerAngle, PtC) → PtA
  28. Polygon({PtA, PtB, PtC}) → triOriginal
  29. PixelsToPoint((xMin, yMin), (xMax, yMax), (800, 500), (180, 180)) → DisplayCenter
  30. Centroid(triOriginal) → CenterTri
  31. Max(Zip[Distance[CenterTri, A], A, {PtA, PtB, PtC}]) → maxLength
  32. Dilate(Rotate[Translate[triOriginal, Vector[CenterTri, DisplayCenter]], randomRotation, DisplayCenter], dilationFactor, DisplayCenter) → triangleQuestion
  33. Vertex(triangleQuestion, 3) → CornerC
  34. Vertex(triangleQuestion, 2) → CornerB
  35. Vertex(triangleQuestion, 1) → CornerA
  36. Zip(Midpoint[J, K], J, TriVertices, K, {TriVertices(2), TriVertices(3), TriVertices(1)}) → midPoints
  37. Zip(Angle[Line[K, J], xAxis], J, TriVertices, K, {TriVertices(2), TriVertices(3), TriVertices(1)}) → sideAngles
  38. Zip(If[A < 90°, 3, If[A < 180°, 2, If[A < 270°, 1, 4]]], A, sideAngles) → cornerNumbers
  39. Zip(numberedCornerBox[boxWidth, boxHeight, A, B], A, midPoints, B, cornerNumbers) → labelBoxes
  40. Sequence(Midpoint[Vertex[labelBoxes(A), 1], Vertex[labelBoxes(A), 3]], A, 1, 3) → homePoints
  41. Element({(-100, -100), PointText1a, PointText2a, PointText3a}, movingInd + 1) → PointMoving
  42. centerBox(boxWidth + 1, (2 * boxHeight), PointPyth3) → region3
  43. centerBox(boxWidth + 1, (2 * boxHeight), PointPyth2) → region2
  44. centerBox(boxWidth + 1, (2 * boxHeight), PointPyth1) → region1
  45. Max({x(PointText1a), x(PointText2a), x(PointText3a)}) → maxX
  46. makeBox((0, 0), boxWidth, boxHeight) → boxTemplate2
  47. If(studentValues(1) ≟ tripleSquaredList(2), tripleSquaredList(2), tripleSquaredList(1)) → solnFirstSq
  48. If(studentValues(1) ≟ tripleSquaredList(2), tripleSquaredList(1), tripleSquaredList(2)) → solnSecondSq
  49. If(cornerAngle ≟ 90°, correctThirdSquared, incorrectThirdSquared) → solnThirdSq
  50. centerBox(boxWidth, boxHeight, PointText3a) → boxText3
  51. Polygon(Zip[PixelsToPoint[Corner[1], Corner[3], Corner[5], A], A, {(pixelsLeft, pixelsTop), (pixelsLeft, y(Corner[5]) - pixelsBottom), Corner[5] - (pixelsRight, pixelsBottom), (x(Corner[5]) - pixelsRight, pixelsTop)}]) → borderBox
  52. centerBox(boxWidth, boxHeight, PointText1a) → boxText1
  53. centerBox(boxWidth, boxHeight, PointText2a) → boxText2
  54. Zip(IsInRegion[PointText1a, A] + IsInRegion[PointText2a, A] + IsInRegion[PointText3a, A], A, {region1, region2, region3}) → countEntries
  55. TableText(row1Soln, row2Soln, row3Soln, "r") → textStepSolution
  56. Angle(CornerA, CornerC, CornerB) → rightAngle
  57. Zip(numberedCornerBox[boxWidth, boxHeight, A, B], A, midPoints, B, cornerNumbers) → staticBoxes
  58. Zip(Centroid[n], n, staticBoxes) → staticCenters
  59. IndexOf(ClosestPoint[staticCenters, (xMax, xMin)], staticCenters) → onboardingIndex
  60. Sequence(Vertex[Pyth1, n], n, 1, 4) → pulseCorners2
  61. centerBox(boxWidth, boxHeight, PointPyth1) → pulsingObject2
  62. IndexOf(ClosestPoint[pythCentersUse, PointMoving], pythCentersUse) → boxInd
  63. IndexOf(maxX, {}) → useIndex
  64. Zip(tripleListText(A), A, {2, 3, 1}) → labelText
  65. IndexOf(true, Zip[IsInRegion[PointMoving, A], A, {region1, region2, region3}]) → destinationIndex
  66. centerBox(boxWidth, boxHeight, destinationLocations(destinationIndex)) → PythHighlight
  67. Zip(Text[("" * A), B, true, true, 0, 0], A, tripleListText, B, {PointText2a, PointText3a, PointText1a}) → movingLabels
  68. centerBox(boxWidth, boxHeight, PointPyth2) → Pyth2Wrong
  69. centerBox(boxWidth, boxHeight, PointPyth1) → Pyth1Wrong
  70. centerBox(boxWidth, boxHeight, PointPyth3) → Pyth3Wrong
  71. Text("\text{Drag to place}", CurrentOnboardingCenter + (7, -0.2), true, true, 1, 0) → onboardingText
  72. Zip(Text[("" * If[Distance[B, C] > 0.5, ("" * A), ""]), B, true, true, 0, 0], A, tripleListText, B, {staticCenters(2), staticCenters(3), staticCenters(1)}, C, {PointText2a, PointText3a, PointText1a}) → staticLabels
  73. If(onboarding ∧ onboardingIndex ≟ 1, pulsing1, 1) → pulsingA
  74. If(onboarding ∧ onboardingIndex ≟ 2, pulsing1, 1) → pulsingB
  75. If(onboarding ∧ onboardingIndex ≟ 3, pulsing1, 1) → pulsingC
  76. centerBox(boxWidth, boxHeight, PointMoving) → movingBackground
  77. TableText(rowSpacing, row1Student, row2Student, "r", 35, 5) → textInputSimplified

AI Drawing Prompt Reference

Create an example demonstrating how to use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle. First, draw a random triangle and label the lengths of its sides. Then, calculate the squares of the two shorter sides and the square of the longest side. Based on the calculation, determine if the triangle is a right triangle and display the relevant mathematical expressions and conclusions next to the diagram. The entire image should include the triangle, side length labels, mathematical expressions, and conclusions to clearly illustrate the application of the converse of the Pythagorean Theorem.
The prompt above was compiled from the original drawing commands and is for reference only.