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点A和点B之间的距离计算

点A和点B之间的距离计算
几何画板及绘图内容加载中，请稍等……
2015-11-12 22:55
三角函数解析几何几何作图
所属专题 CCSS Support Math I Unit 6
①本范例演示如何计算二维坐标系中两点A和B之间的距离；②涉及的核心数学概念包括坐标几何、勾股定理和距离公式；③作图要点——从下方『构造摘要』提炼用到了哪些 GeoGebra 命令/对象，如SelectedElement, Polygon, Intersect, Point, Vector, LaTeX, Segment等；④适合高中数学学生学习和理解解析几何中的基本计算方法。

GeoGebra 命令及参数

  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