Pythagorean Theorem: Calculating Hypotenuse in Right Triangles

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2024-09-26 21:25

In collections Satz des Pythagoras - Einführung

This interactive GeoGebra material helps students master the practical application of the Pythagorean theorem. It includes: ①calculating hypotenuse length from given legs (a=5cm, b=12cm or a=15cm, b=8cm); ②understanding various forms of the theorem (a²+b²=c², e²=f²+g², etc.); ③explaining why different symbol notations are equivalent. The construction uses GeoGebra commands including Textfield, Point, Segment, Polygon, Angle, Distance, and Rotate to draw right triangles and dynamically display calculations. Suitable for Grade 7-8 students, ideal for classroom practice or self-review, developing students' ability to correctly write and apply the Pythagorean theorem.

GeoGebra Commands & Parameters

Textfield(input12) → Eingabefeld12
SelectedIndex(DropDown21) → input21
Element({4, 5, 6, 6.1, 7, 8, 9, 10, 11, 12, 13, 15, 16, 18, 19}, step2) → step
If(step ≟ 16, (11.4, -3.4), If[step ≥ 17, A1Ende, A1Start]) → A1
SelectedIndex(DropDown3) → input3
SelectedIndex(DropDown1) → input1
SelectedIndex(DropDown5) → input5
SelectedIndex(DropDown6) → input6
SelectedIndex(DropDown7) → input7
Vector(C1, A1) → v2_{1}
Vector(C1, B1) → v1_{1}
Sequence(O1 + (m * (v2_{1} + (0.6, 0))), m, 0, 3) → listO
Point(Segment[listO(4), O2]) → C4
Textfield(input9) → Eingabefeld9
Point(Segment[listO(3), O3]) → C3
Point(Segment[listO(2), O4]) → C2
Point(Segment[B1, C1]) → S1
Point(Segment[A1, C1]) → S2
Textfield(input2) → Eingabefeld2
Point(Line[V1, xAxis]) → V2
Segment(B1, A1) → s3
Segment(B1, S1) → s1
Segment(A1, S2) → s2
Angle(Polygon[C1, A1, B1]) → wa3, wa2, wa1
Polygon(A2, B2, C2) → d2, t3, t1, t2
Polygon(B3, C3, A3) → d3, u1, u2, u3
Polygon(A4, B4, C4) → d4, v3, v1, v2
Angle(d2) → wb2, wb1, wb3
Angle(d3) → wc1, wc3, wc2
Angle(d4) → wd2, wd1, wd3
Zip(Midpoint[A, B] + (0.2 * Vector[C, Midpoint[A, B]]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listM3
Zip(Midpoint[B, C] + (0.15 * Vector[A, C]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listM1
Zip(Midpoint[A, C] + (0.09 * Vector[B, C]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listM2
Zip(B + (0.19 * Vector[B, C + (1 / 2 * (A - C))]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listMB
Angle(A1, B1, B2) → wa4
Angle(A2, B2, B3) → wb4
Angle(A3, B3, B4) → wc4
Angle(A4, B4, B1) → wd4
Point(Line[L1, xAxis]) → L2
Zip(A + (0.18 * Vector[C, B]) + (0.08 * Vector[C, A]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listMD
Point(Line[L3, xAxis]) → L4
Distance(L3, L4) → lupesize2
Rotate(R1a, wa1, Q1) → R1b
Angle(R1a, Q1, R1b) → va1
Rotate(R1a, wa1 + wa3, Q1) → R1c
Rotate(R1c, wa2, Q1) → R1d
Angle(R1c, Q1, R1d) → va2
Angle(R1b, Q1, R1c) → va3
Rotate(R2a, wa1, Q2) → R2b
Angle(R2a, Q2, R2b) → vb1
Rotate(R2b, wa3, Q2) → R2c
Rotate(R2c, wa2, Q2) → R2d
Angle(R2b, Q2, R2c) → vb3
Angle(R2c, Q2, R2d) → vb2
Rotate(R3a, (-wa2), Q3) → R3b
Rotate(R3b, (-wa3), Q3) → R3c
Rotate(R3c, (-wa1), Q3) → R3d
Angle(R3d, Q3, R3c) → vc1
Angle(R3b, Q3, R3a) → vc2
Angle(R3c, Q3, R3b) → vc3
Rotate(R4a, wa1, Q4) → R4b
Rotate(R4b, wa3, Q4) → R4c
Rotate(R4c, wa2, Q4) → R4d
Angle(R4a, Q4, R4b) → vd1
Angle(R4b, Q4, R4c) → vd3
Angle(R4c, Q4, R4d) → vd2
Textfield(input4) → Eingabefeld4
Polygon(A1, B1, C1) → d1, c1, a1, b1
Textfield(input14) → Eingabefeld14
Textfield(input15) → Eingabefeld15
Textfield(input16) → Eingabefeld16
Textfield(input17) → Eingabefeld17
Textfield(input18) → Eingabefeld18
Textfield(input19) → Eingabefeld19
LaTeX(step, true, true) → Text12
Polygon(DFlink3, DFlink1, DFlink2) → dflink, dflink3, dflink1, dflink2
Angle(dflink) → wflink1, wflink2, wflink3
Textfield(input20) → Eingabefeld20
CountIf(m ≟ true, m, {selected1, selected2, selected3, selected4}) → numselected
Textfield(input10) → Eingabefeld10
Textfield(input11) → Eingabefeld11
Textfield(input13) → Eingabefeld13
If(path1, Pos0 + (1.3, -1.9), Pos0 + (1.3, -3.3)) → Pos9
If(path1, Pos0 + (3.45, -2.02), Pos0 + (3.45, -3.6)) → Pos10
If(path1, Pos0 + (5.75, -2.02), Pos0 + (5.75, -3.6)) → Pos11
Element(Join[{Sequence[Posi0 + (m * (0, -1.22)) + (1.1, -1), m, 0, 2], {Posi0 + (2.4, -5.2)}}], If[correct18, 4, If[correct16 ∧ correct17, 3, If[correct14 ∧ correct15, 2, 1]]]) → PosiSol
Angle(A1, B1, B2) → wa4punkt
Angle(A2, B2, B3) → wb4punkt
Angle(A3, B3, B4) → wc4punkt
Angle(A4, B4, B1) → wd4punkt
Zip(A + (0.25 * Vector[A, C + (2 / 5 * (B - C))]), A, {A1, A2, A3, A4}, B, {B1, B2, B3, B4}, C, {C1, C2, C3, C4}) → listMA
If(path1, Pos0 + (4.9, -3.3), Pos0 + (3.7, -1.9)) → Pos12
Element(Sequence[Pos0 + (2, -3.1) + (m * (0, -1.3)), m, 0, 2], If[path1, If[correct12, 3, If[correct9 ∧ correct10 ∧ correct11, 2, 1]], If[correct9 ∧ correct10 ∧ correct11, 3, If[correct12, 2, 1]]]) → PosSol4
If(path1, If[correct12, 3, If[correct9 ∧ correct10 ∧ correct11, 2, 1]], If[correct9 ∧ correct10 ∧ correct11, 3, If[correct12, 2, 1]]) → solPos
Point(Line[Py1, xAxis]) → Py2

AI Drawing Prompt Reference

Create a right-triangle geometry diagram for learning the Pythagorean theorem: first draw a right triangle by constructing two perpendicular legs and the connecting hypotenuse, keeping clear vertex-to-edge relationships. Mark the three interior angles at the triangle’s vertices (the angles formed by each pair of connected sides). Add several auxiliary polygons that share the same key vertex positions and connect to the original triangle to illustrate corresponding configurations. Then perform a rotation around one vertex using the marked angle as the starting direction to generate a new rotated position, and label the resulting angle to show how the angle measure relates before and after rotation.

The prompt above was compiled from the original drawing commands and is for reference only.