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Applying and Verifying the Pythagorean Theorem

Applying and Verifying the Pythagorean Theorem
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2024-09-26 21:26
TriangleAngleGeometric ConstructionSlider Animation
In collections Satz des Pythagoras - Einführung
This example demonstrates how to use the Pythagorean theorem to determine if a triangle is right-angled. The core math concepts involved are the Pythagorean theorem and its converse. GeoGebra commands and objects used in construction include Textfield, Midpoint, Point, Segment, Join, Sequence, IndexOf, Rotate, Angle, If, etc. It is suitable for 7th and 8th-grade students to learn and practice.

GeoGebra Commands & Parameters

  1. Textfield(input6a) → Eingabefeld6
  2. Textfield(input2a) → Eingabefeld2
  3. Textfield(input1a) → Eingabefeld1
  4. Midpoint(V1, V4) → P3Start
  5. Textfield(input7) → Eingabefeld7
  6. Textfield(input5) → Eingabefeld5
  7. Textfield(input4) → Eingabefeld4
  8. Textfield(input3) → Eingabefeld3
  9. SelectedIndex(DropDown8) → input8
  10. Point(Segment[Dilate[V4, 0.5, Midpoint[V1, V4]], Dilate[V1, 0, Midpoint[V1, V4]]]) → P3
  11. Point(Segment[Dilate[V1, 0.5, Midpoint[V2, V1]], Dilate[V2, 0.5, Midpoint[V2, V1]]]) → P1
  12. Point(Segment[Dilate[V3, 0.5, Midpoint[V3, V2]], Dilate[V2, 0, Midpoint[V3, V2]]]) → P2
  13. Element({P2, P3, P1}, indexC) → PA
  14. Element({P1, P2, P3}, indexC) → PC
  15. Segment(PA, PC) → sb
  16. Element({P3, P1, P2}, indexC) → PB
  17. Segment(PB, PC) → sa
  18. Join({Sequence[Text[UnicodeToLetter[65 + k], Midpoint[Corner[4] + (RasterPixMat * ((k * RasterStepXY), RasterMaxY)), Corner[4] + (RasterPixMat * (((k + 1) * RasterStepXY), RasterMaxY))], false, true, 0, -1], k, 0, RasterMaxX / RasterStepXY - 1], Sequence[Text[("" * k), Midpoint[Corner[4] + (RasterPixMat * (RasterMaxX, ((k - 1) * RasterStepXY))), Corner[4] + (RasterPixMat * (RasterMaxX, (k * RasterStepXY)))], false, true, 1, 0], k, 1, RasterMaxY / RasterStepXY]}) → RasterBeschriftung
  19. Segment(P1, P2) → s3
  20. Segment(P2, P3) → s1
  21. Segment(P3, P1) → s2
  22. Element({2, 3, 1}, indexC) → indexA
  23. Element({3, 1, 2}, indexC) → indexB
  24. Segment(PA, PB) → sc
  25. Midpoint(s2) → M2
  26. Midpoint(s3) → M3
  27. Element(Sort[{colouroldgreen, colourblue, colourorange}, {indexA, indexB, indexC}], 1) → colour1
  28. Midpoint(s1) → M1
  29. Element(Sort[{colouroldgreen, colourblue, colourorange}, {indexA, indexB, indexC}], 2) → colour2
  30. Element(Sort[{colouroldgreen, colourblue, colourorange}, {indexA, indexB, indexC}], 3) → colour3
  31. Join({Sequence[Segment[Corner[4] + (RasterPixMat * ((k * RasterStepXY), 0)), Corner[4] + (RasterPixMat * ((k * RasterStepXY), RasterMaxY))], k, 0, RasterMaxX / RasterStepXY], Sequence[Segment[Corner[4] + (RasterPixMat * (0, (k * RasterStepXY))), Corner[4] + (RasterPixMat * (RasterMaxX, (k * RasterStepXY)))], k, 0, RasterMaxY / RasterStepXY]}) → RasterStrecken
  32. Point(Line[PosSol1a, yAxis]) → PosSol1b
  33. Point(Line[PosSol1a, yAxis]) → PosSol1c
  34. Point(Line[PosSol1d, yAxis]) → PosSol1e
  35. Sequence(Sequence[Corner[4] + (RasterPixMat * ((k * RasterStepXY), (m * RasterStepXY))), k, 0, RasterMaxX / RasterStepXY], m, 0, RasterMaxY / RasterStepXY) → RasterPunkte, {{(-17.9, 12.100000000000033), (-16.89749373433584, 12.100000000000033), (-15.89498746867168, 12.100000000000033), (-14.89248120300752, 12.100000000000033), (-13.889974937343359, 12.100000000000033), (-12.8874686716792, 12.100000000000033), (-11.88496240601504, 12.100000000000033), (-10.882456140350879, 12.100000000000033), (-9.87994987468672, 12.100000000000033), (-8.87744360902256, 12.100000000000033), (-7.8749373433584005, 12.100000000000033), (-6.872431077694241, 12.100000000000033), (-5.8699248120300815, 12.100000000000033), (-4.86741854636592, 12.100000000000033), (-3.864912280701761, 12.100000000000033), (-2.8624060150376014, 12.100000000000033), (-1.8598997493734402, 12.100000000000033)}, {(-17.9, 11.09665551839468), (-16.89749373433584, 11.09665551839468), (-15.89498746867168, 11.09665551839468), (-14.89248120300752, 11.09665551839468), (-13.889974937343359, 11.09665551839468), (-12.8874686716792, 11.09665551839468), (-11.88496240601504, 11.09665551839468), (-10.882456140350879, 11.09665551839468), (-9.87994987468672, 11.09665551839468), (-8.87744360902256, 11.09665551839468), (-7.8749373433584005, 11.09665551839468), (-6.872431077694241, 11.09665551839468), (-5.8699248120300815, 11.09665551839468), (-4.86741854636592, 11.09665551839468), (-3.864912280701761, 11.09665551839468), (-2.8624060150376014, 11.09665551839468), (-1.8598997493734402, 11.09665551839468)}, {(-17.9, 10.093311036789327), (-16.89749373433584, 10.093311036789327), (-15.89498746867168, 10.093311036789327), (-14.89248120300752, 10.093311036789327), (-13.889974937343359, 10.093311036789327), (-12.8874686716792, 10.093311036789327), (-11.88496240601504, 10.093311036789327), (-10.882456140350879, 10.093311036789327), (-9.87994987468672, 10.093311036789327), (-8.87744360902256, 10.093311036789327), (-7.8749373433584005, 10.093311036789327), (-6.872431077694241, 10.093311036789327), (-5.8699248120300815, 10.093311036789327), (-4.86741854636592, 10.093311036789327), (-3.864912280701761, 10.093311036789327), (-2.8624060150376014, 10.093311036789327), (-1.8598997493734402, 10.093311036789327)}, {(-17.9, 9.089966555183972), (-16.89749373433584, 9.089966555183972), (-15.89498746867168, 9.089966555183972), (-14.89248120300752, 9.089966555183972), (-13.889974937343359, 9.089966555183972), (-12.8874686716792, 9.089966555183972), (-11.88496240601504, 9.089966555183972), (-10.882456140350879, 9.089966555183972), (-9.87994987468672, 9.089966555183972), (-8.87744360902256, 9.089966555183972), (-7.8749373433584005, 9.089966555183972), (-6.872431077694241, 9.089966555183972), (-5.8699248120300815, 9.089966555183972), (-4.86741854636592, 9.089966555183972), (-3.864912280701761, 9.089966555183972), (-2.8624060150376014, 9.089966555183972), (-1.8598997493734402, 9.089966555183972)}, {(-17.9, 8.086622073578619), (-16.89749373433584, 8.086622073578619), (-15.89498746867168, 8.086622073578619), (-14.89248120300752, 8.086622073578619), (-13.889974937343359, 8.086622073578619), (-12.8874686716792, 8.086622073578619), (-11.88496240601504, 8.086622073578619), (-10.882456140350879, 8.086622073578619), (-9.87994987468672, 8.086622073578619), (-8.87744360902256, 8.086622073578619), (-7.8749373433584005, 8.086622073578619), (-6.872431077694241, 8.086622073578619), (-5.8699248120300815, 8.086622073578619), (-4.86741854636592, 8.086622073578619), (-3.864912280701761, 8.086622073578619), (-2.8624060150376014, 8.086622073578619), (-1.8598997493734402, 8.086622073578619)}, {(-17.9, 7.083277591973265), (-16.89749373433584, 7.083277591973265), (-15.89498746867168, 7.083277591973265), (-14.89248120300752, 7.083277591973265), (-13.889974937343359, 7.083277591973265), (-12.8874686716792, 7.083277591973265), (-11.88496240601504, 7.083277591973265), (-10.882456140350879, 7.083277591973265), (-9.87994987468672, 7.083277591973265), (-8.87744360902256, 7.083277591973265), (-7.8749373433584005, 7.083277591973265), (-6.872431077694241, 7.083277591973265), (-5.8699248120300815, 7.083277591973265), (-4.86741854636592, 7.083277591973265), (-3.864912280701761, 7.083277591973265), (-2.8624060150376014, 7.083277591973265), (-1.8598997493734402, 7.083277591973265)}, {(-17.9, 6.079933110367912), (-16.89749373433584, 6.079933110367912), (-15.89498746867168, 6.079933110367912), (-14.89248120300752, 6.079933110367912), (-13.889974937343359, 6.079933110367912), (-12.8874686716792, 6.079933110367912), (-11.88496240601504, 6.079933110367912), (-10.882456140350879, 6.079933110367912), (-9.87994987468672, 6.079933110367912), (-8.87744360902256, 6.079933110367912), (-7.8749373433584005, 6.079933110367912), (-6.872431077694241, 6.079933110367912), (-5.8699248120300815, 6.079933110367912), (-4.86741854636592, 6.079933110367912), (-3.864912280701761, 6.079933110367912), (-2.8624060150376014, 6.079933110367912), (-1.8598997493734402, 6.079933110367912)}, {(-17.9, 5.076588628762558), (-16.89749373433584, 5.076588628762558), (-15.89498746867168, 5.076588628762558), (-14.89248120300752, 5.076588628762558), (-13.889974937343359, 5.076588628762558), (-12.8874686716792, 5.076588628762558), (-11.88496240601504, 5.076588628762558), (-10.882456140350879, 5.076588628762558), (-9.87994987468672, 5.076588628762558), (-8.87744360902256, 5.076588628762558), (-7.8749373433584005, 5.076588628762558), (-6.872431077694241, 5.076588628762558), (-5.8699248120300815, 5.076588628762558), (-4.86741854636592, 5.076588628762558), (-3.864912280701761, 5.076588628762558), (-2.8624060150376014, 5.076588628762558), (-1.8598997493734402, 5.076588628762558)}, {(-17.9, 4.0732441471572045), (-16.89749373433584, 4.0732441471572045), (-15.89498746867168, 4.0732441471572045), (-14.89248120300752, 4.0732441471572045), (-13.889974937343359, 4.0732441471572045), (-12.8874686716792, 4.0732441471572045), (-11.88496240601504, 4.0732441471572045), (-10.882456140350879, 4.0732441471572045), (-9.87994987468672, 4.0732441471572045), (-8.87744360902256, 4.0732441471572045), (-7.8749373433584005, 4.0732441471572045), (-6.872431077694241, 4.0732441471572045), (-5.8699248120300815, 4.0732441471572045), (-4.86741854636592, 4.0732441471572045), (-3.864912280701761, 4.0732441471572045), (-2.8624060150376014, 4.0732441471572045), (-1.8598997493734402, 4.0732441471572045)}, {(-17.9, 3.0698996655518513), (-16.89749373433584, 3.0698996655518513), (-15.89498746867168, 3.0698996655518513), (-14.89248120300752, 3.0698996655518513), (-13.889974937343359, 3.0698996655518513), (-12.8874686716792, 3.0698996655518513), (-11.88496240601504, 3.0698996655518513), (-10.882456140350879, 3.0698996655518513), (-9.87994987468672, 3.0698996655518513), (-8.87744360902256, 3.0698996655518513), (-7.8749373433584005, 3.0698996655518513), (-6.872431077694241, 3.0698996655518513), (-5.8699248120300815, 3.0698996655518513), (-4.86741854636592, 3.0698996655518513), (-3.864912280701761, 3.0698996655518513), (-2.8624060150376014, 3.0698996655518513), (-1.8598997493734402, 3.0698996655518513)}, {(-17.9, 2.0665551839464964), (-16.89749373433584, 2.0665551839464964), (-15.89498746867168, 2.0665551839464964), (-14.89248120300752, 2.0665551839464964), (-13.889974937343359, 2.0665551839464964), (-12.8874686716792, 2.0665551839464964), (-11.88496240601504, 2.0665551839464964), (-10.882456140350879, 2.0665551839464964), (-9.87994987468672, 2.0665551839464964), (-8.87744360902256, 2.0665551839464964), (-7.8749373433584005, 2.0665551839464964), (-6.872431077694241, 2.0665551839464964), (-5.8699248120300815, 2.0665551839464964), (-4.86741854636592, 2.0665551839464964), (-3.864912280701761, 2.0665551839464964), (-2.8624060150376014, 2.0665551839464964), (-1.8598997493734402, 2.0665551839464964)}, {(-17.9, 1.0632107023411432), (-16.89749373433584, 1.0632107023411432), (-15.89498746867168, 1.0632107023411432), (-14.89248120300752, 1.0632107023411432), (-13.889974937343359, 1.0632107023411432), (-12.8874686716792, 1.0632107023411432), (-11.88496240601504, 1.0632107023411432), (-10.882456140350879, 1.0632107023411432), (-9.87994987468672, 1.0632107023411432), (-8.87744360902256, 1.0632107023411432), (-7.8749373433584005, 1.0632107023411432), (-6.872431077694241, 1.0632107023411432), (-5.8699248120300815, 1.0632107023411432), (-4.86741854636592, 1.0632107023411432), (-3.864912280701761, 1.0632107023411432), (-2.8624060150376014, 1.0632107023411432), (-1.8598997493734402, 1.0632107023411432)}, {(-17.9, 0.059866220735790066), (-16.89749373433584, 0.059866220735790066), (-15.89498746867168, 0.059866220735790066), (-14.89248120300752, 0.059866220735790066), (-13.889974937343359, 0.059866220735790066), (-12.8874686716792, 0.059866220735790066), (-11.88496240601504, 0.059866220735790066), (-10.882456140350879, 0.059866220735790066), (-9.87994987468672, 0.059866220735790066), (-8.87744360902256, 0.059866220735790066), (-7.8749373433584005, 0.059866220735790066), (-6.872431077694241, 0.059866220735790066), (-5.8699248120300815, 0.059866220735790066), (-4.86741854636592, 0.059866220735790066), (-3.864912280701761, 0.059866220735790066), (-2.8624060150376014, 0.059866220735790066), (-1.8598997493734402, 0.059866220735790066)}}
  36. IndexOf(Max[{s1, s2, s3}], Zip[Length[a], a, {s1, s2, s3}]) → indexCaktuell
  37. Join({Sequence[Segment[Corner[4] + (RasterPixMat * ((k * RasterStepXY), 0)), Corner[4] + (RasterPixMat * ((k * RasterStepXY), RasterMaxY))], k, RasterMaxX / RasterStepXY, RasterMaxX / RasterStepXY], Sequence[Segment[Corner[4] + (RasterPixMat * (0, (k * RasterStepXY))), Corner[4] + (RasterPixMat * (RasterMaxX, (k * RasterStepXY)))], k, RasterMaxY / RasterStepXY, RasterMaxY / RasterStepXY]}) → RasterAussen
  38. Rotate(Segment[P1 + (ArrowDistance, 0), P1 + (ArrowDistance + ArrowLength, 0)], 0° + ((1 - 1) * (360°) / 4), P1) → arrowP1b
  39. Rotate(Segment[P1 + (ArrowDistance, 0), P1 + (ArrowDistance + ArrowLength, 0)], 0° + ((3 - 1) * (360°) / 4), P1) → arrowP1a
  40. Rotate(Segment[P2 + (ArrowDistance, 0), P2 + (ArrowDistance + ArrowLength, 0)], ArrowAngle + ((1 - 1) * (360°) / ArrowNumber), P2) → arrowP2a
  41. Rotate(Segment[P2 + (ArrowDistance, 0), P2 + (ArrowDistance + ArrowLength, 0)], ArrowAngle + ((2 - 1) * (360°) / ArrowNumber), P2) → arrowP2b
  42. Rotate(Segment[P3 + (ArrowDistance, 0), P3 + (ArrowDistance + ArrowLength, 0)], ArrowAngle + ((1 - 1) * (360°) / ArrowNumber), P3) → arrowP3a
  43. Rotate(Segment[P3 + (ArrowDistance, 0), P3 + (ArrowDistance + ArrowLength, 0)], ArrowAngle + ((2 - 1) * (360°) / ArrowNumber), P3) → arrowP3b
  44. Angle(P3, P2, P1) → w2
  45. Angle(P2, P1, P3) → w1
  46. Angle(P1, P3, P2) → w3
  47. Midpoint(V1, V2) → P1Start
  48. Midpoint(Midpoint[V2, V3], V3) → P2Start
  49. If(step ≟ maxstep, Ayla2, Ayla1) → Aylaa
  50. Circle(Midpoint[Corner[TextBlitz, 1], Corner[TextBlitz, 3]], (0.4 * Distance[Corner[TextBlitz, 1], Corner[TextBlitz, 3]])) → kreisBlitz

AI Drawing Prompt Reference

Create a triangle on a coordinate plane by defining three points and joining them with three segments to form the closed triangular boundary. Construct the midpoints of the three sides and use these midpoint points together with the triangle’s vertices to guide the geometric relationships inside the triangle. Then mark and label the three interior angles at the triangle’s vertices, so each labeled angle corresponds to one corner of the triangle.
The prompt above was compiled from the original drawing commands and is for reference only.