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Distance Between Points A and B

Distance Between Points A and B
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2015-11-12 22:55
TrigonometryCoordinate GeometryGeometric Construction
In collections CCSS Support Math I Unit 6
This example demonstrates how to calculate the distance between two points A and B in a 2D coordinate system. Key math concepts include coordinate geometry, Pythagorean theorem, and distance formula. Construction highlights involve GeoGebra commands like SelectedElement, Polygon, Intersect, Point, Vector, LaTeX, Segment, etc. Suitable for high school students learning basic computational methods in analytical geometry.

GeoGebra Commands & Parameters

  1. SelectedElement(valueE) → vare
  2. SelectedElement(valueB) → varb
  3. SelectedElement(valueD) → vard
  4. SelectedElement(valueA) → vara
  5. SelectedElement(valueB) → XCA
  6. SelectedElement(valueA) → XCB
  7. Polygon(A', B', C', D) → poly1, a, b, c, d
  8. Polygon(E, F, G, H) → poly2, e, f, g, h
  9. Polygon(I, J, K, L) → poly3, i', j, k, l
  10. Intersect(xAxis, yAxis) → M
  11. Point(xAxis) → N
  12. Vector(M, N) → u
  13. Point(yAxis) → O
  14. Vector(M, O) → v
  15. Point(yAxis) → R
  16. Vector(M, R) → w
  17. Point(xAxis) → S
  18. Vector(M, S) → m
  19. Polygon(T, U, V, W) → poly4, t, u_1, v_1, w_1
  20. LaTeX("\text{Find the distance between points \textcolor{#000099}{A} and \textcolor{#000099}{B}.}") → Question1
  21. LaTeX("\text{Draw a right triangle using the given points.}") → step1
  22. LaTeX("\text{Either of these coordinates can be used for the third point in the right }") → step4_1
  23. LaTeX("\text{Find the length of the hypotenuse of the right triangle.}") → step6
  24. LaTeX("\text{Use the Pythagorean Theorem to find the length of the hypotenuse }") → step7_1
  25. LaTeX("\text{of the right triangle.In this triangle,the hypotenuse is the distance}") → step7_2
  26. LaTeX("\text{hypotenuse.}") → step8_3
  27. LaTeX("\text{Recall that the lengths of the legs can be found through subtraction,}") → step9_1
  28. LaTeX("\text{using the coordinates of the legs’ endpoints.For the horizontal leg, }") → step9_2
  29. LaTeX("\text{on the graph.}") → step9_6
  30. LaTeX("\text{Pythagorean Theorem}") → step11_2
  31. LaTeX("\text{Add.}") → step14_2
  32. LaTeX("\text{Take the positive square root of}") → step15_2
  33. LaTeX("\text{both sides of the equation.}") → step15_3
  34. LaTeX("\text{Simplify.}") → step16_2
  35. LaTeX("\text{same \textcolor{#000099}{y}-coordinate as \textcolor{#000099}{B},or with the same \textcolor{#000099}{x}-coordinate as \textcolor{#000099}{B} and the }") → step2_2
  36. LaTeX("\text{same \textcolor{#000099}{y}-coordinate as \textcolor{#000099}{A}.}") → step2_3
  37. LaTeX("\text{Plot another point either with the same \textcolor{#000099}{x}-coordinate as \textcolor{#000099}{A} and the }") → step2_1
  38. LaTeX("\text{\textcolor{#000099}{A,B},and \textcolor{#000099}{C} as vertices,as shown. }") → step5_2
  39. LaTeX("\text{between points \textcolor{#000099}{A} and \textcolor{#000099}{B}.}") → step7_3
  40. LaTeX("\text{can be set up and solved for \textcolor{#000099}{c}.}") → step10_3
  41. Segment(A, C) → n
  42. Segment(C, B) → p
  43. Segment(A, B) → q
  44. LaTeX((((("\text{\textcolor{#000099}{(" * varb) * ",") * vare) * ")}.Therefore,the point with the same \textcolor{#000099}{x}-coordinate as \textcolor{#000099}{A} and}")) → step3_2
  45. LaTeX((((("\text{The coordinates of point \textcolor{#000099}{A} are \textcolor{#000099}{(" * vara) * ", ") * vard) * ")},and those of point \textcolor{#000099}{B} are}")) → step3_1
  46. LaTeX((((("\text{triangle.Let’s use \textcolor{#000099}{(" * vara) * ",") * vare) * ")}.}")) → step4_2
  47. LaTeX((((("\text{Plot a point at \textcolor{#000099}{(" * vara) * ",") * vare) * ")} and label it \textcolor{#000099}{C}.Then draw a right triangle with}")) → step5_1
  48. LaTeX("\text{The Pythagorean Theorem is \textcolor{#000099}{a^{2}+b^{2}=c^{2}},where \textcolor{#000099}{a} and \textcolor{#000099}{b} are the}") → step8_1
  49. LaTeX("\text{lengths of the legs of a right triangle and \textcolor{#000099}{c} is the length of the}") → step8_2
  50. LaTeX((((("\text{\textcolor{#000099}{(" * vara) * ",") * vare) * ")}.You can also count the number of units for each leg as shown}")) → step9_5
  51. LaTeX((("\text{The length of the hypotenuse,\textcolor{#000099}{c},is \textcolor{#000099}{" * result) * "} units.Therefore,the distance}")) → step17_1
  52. LaTeX((("\text{between points \textcolor{#000099}{A} and \textcolor{#000099}{B} is \textcolor{#000099}{" * result) * "} units.}")) → step17_2
  53. LaTeX((((("\text{Square \textcolor{#000099}{" * val1) * "} and \textcolor{#000099}{") * val2) * "}.}")) → step13_2
  54. LaTeX("\text{\textcolor{#000099}{a^2+b^2=c^2.}}") → data2_3
  55. LaTeX("\text{right triangle(\textcolor{#000099}{c}) to the lengths of its legs (\textcolor{#000099}{a} and \textcolor{#000099}{b}).The theorem states that}") → data2_2
  56. LaTeX((((("\text{the same \textcolor{#000099}{y}-coordinate as \textcolor{#000099}{B} is at \textcolor{#000099}{(" * vara) * ",") * vare) * ")},and the point with the}")) → step3_3
  57. LaTeX((((("\text{same \textcolor{#000099}{x}-coordinate as \textcolor{#000099}{B} and the same \textcolor{#000099}{y}-coordinate as \textcolor{#000099}{A} is at \textcolor{#000099}{(" * varb) * ",") * vard) * ")}.}")) → step3_4
  58. LaTeX((((((((("\text{\textcolor{#000099}{a},subtract the \textcolor{#000099}{x}-values of the coordinates \textcolor{#000099}{(" * varb) * ",") * vare) * ")} and \textcolor{#000099}{(") * vara) * ",") * vare) * ")}.For}")) → step9_3
  59. LaTeX((((("\text{the vertical leg,\textcolor{#000099}{b},subtract the \textcolor{#000099}{y}-values of the coordinates \textcolor{#000099}{(" * vara) * ",") * vard) * ")} and}")) → step9_4
  60. LaTeX((((((("\text{In this case,the lengths of the legs are \textcolor{#000099}{" * varb) * "\;–\;(") * vara) * ")=") * val1) * "} units and}")) → step10_1
  61. LaTeX((((((((((("\text{\textcolor{#000099}{" * vard) * "\;–\;(") * vare) * ")=") * val2) * "} units,respectively.Therefore,the equation \textcolor{#000099}{") * val1) * "^{2}+") * val2) * "^{2}=c^{2}}}")) → step10_2
  62. LaTeX("\text{\textcolor{#000099}{a^{2}+b^{2}}") → step11_1
  63. LaTeX("\text{\textcolor{#000099}{=c^{2}}") → step11_3
  64. LaTeX((((("\text{\textcolor{#000099}{(" * val1) * ")^{2}+(") * val2) * ")^{2}}}")) → step12_1
  65. LaTeX("\text{\textcolor{#000099}{=c^{2}}") → step11_4
  66. LaTeX((((("\text{\textcolor{#000099}{" * square1) * "+") * square2) * "}")) → step13_1
  67. LaTeX("\text{\textcolor{#000099}{=c^{2}}") → step11_5
  68. LaTeX((("\text{\textcolor{#000099}{" * total) * "}")) → step14_1
  69. LaTeX("\text{\textcolor{#000099}{=c^{2}}") → step11_6
  70. LaTeX((("\text{\textcolor{#000099}{\sqrt{" * total) * "}}")) → step15_1
  71. LaTeX("\text{\textcolor{#000099}{=\sqrt{c^{2}}") → step11_7
  72. LaTeX((("\text{\textcolor{#000099}{" * result) * "}")) → step16_1
  73. LaTeX("\text{\textcolor{#000099}{=c}}") → step11_8
  74. LaTeX((((("\text{Substitute \textcolor{#000099}{" * val1) * "} for \textcolor{#000099}{a} and \textcolor{#000099}{") * val2) * "} for \textcolor{#000099}{b}.}")) → step12_2
  75. LaTeX("\text{\mathbf{Pythagorean\;Theorem} a theorem that relates the length of the hypotenuse of a}") → data2_1
  76. Segment(Z, A_1) → i
  77. Segment(B_1, C_1) → r
  78. Segment(D_1, E_1) → s
  79. Segment(F_1, G_1) → a_1
  80. Segment(H_1, I_1) → b_1
  81. Segment(J_1, O_1) → c_1
  82. Segment(K_1, P_1) → d_1
  83. Segment(L_1, Q_1) → e_1
  84. Segment(M_1, R_1) → f_1
  85. Segment(N_1, S_1) → g_1
  86. LaTeX("\text{\mathbf{Hypotenuse} the side opposite the vertex of the \textcolor{#000099}{90º} angle in a right triangle.}") → data1
  87. LaTeX("\text{X coordinate of A:}") → DKT1
  88. LaTeX("\text{Y coordinate of A:}") → DKTr1
  89. LaTeX("\text{X coordinate of B:}") → DKT2
  90. LaTeX("\text{Y coordinate of B:}") → DKTr2