3.10 坐标平面上的距离公式

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  坐标平面中的距离公式

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  Distance Formula in the Coordinate Plane
  ::坐标平面中的距离公式

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  The distance between two points $\begin{array}{r}(x_1,y_1)\end{array}$ and $\begin{array}{r}(x_2,y_2)\end{array}$ can be defined as $\begin{array}{r}d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}\end{array}$ . This is called the distance formula . Remember that distances are always positive!
  ::两个点( x1,y1) 和 (x2,y2) 之间的距离可以定义为 d= (x2 - x1) 2+ (y2 -y1) 2。 这被称为距离公式。 记住距离总是正数 !

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  What if you were given the coordinates of two points? How could you find how far apart these two points are?
  ::假若你们获得两点坐标的坐标,你们怎能发现这两点相距甚远呢?

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  Examples
  ::实例

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  Example 1
  ::例1

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  Find the distance between (-2, -3) and (3, 9).
  ::查找( 2, 2, 3) 和( 3, 9) 之间的距离 。

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  Use the distance formula, plug in the points, and simplify.
  ::使用距离公式,插入点并简化。

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  $$\begin{array}{rl}d& =\sqrt{(3-(-2){)}^2+(9-(-3){)}^2}\\ & =\sqrt{(5{)}^2+(12{)}^2}\\ & =\sqrt{25+144}\\ & =\sqrt{169}=13units\end{array}$$

  
  ::d=(3-(-2))2+(9-(-3))2+(9-(3))2=(5)2+(12)2=25+144=169=13单位

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  Example 2
  ::例2

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  Find the distance between (12, 26) and (8, 7).
  ::查找(12、26)和(8、7)之间的距离。

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  Use the distance formula, plug in the points, and simplify.
  ::使用距离公式,插入点并简化。

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  $$\begin{array}{rl}d& =\sqrt{(8-12{)}^2+(7-26{)}^2}\\ & =\sqrt{(-4{)}^2+(-19{)}^2}\\ & =\sqrt{16+361}\\ & =\sqrt{377}\approx 19.42units\end{array}$$

  
  ::d=(8-12)2+(7-26)2=(-4)2+(19)2+(19)2=16+361=377*19.42单位

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  Example 3
  ::例3

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  Find the distance between (4, -2) and (-10, 3).
  ::查找(4, 2)和(10, 3)之间的距离。

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  Plug in (4, -2) for $\begin{array}{r}(x_1,y_1)\end{array}$ and (-10, 3) for $\begin{array}{r}(x_2,y_2)\end{array}$ and simplify.
  ::插插(4, 2),(x1,y1)和(10, 3),(x2,y2)和简化。

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  $$\begin{array}{rl}d& =\sqrt{(-10-4{)}^2+(3+2{)}^2}\\ & =\sqrt{(-14{)}^2+(5{)}^2}\\ & =\sqrt{196+25}\\ & =\sqrt{221}\approx 14.87units\end{array}$$

  
  ::d=(-10-4)2+(3+2)2=(-)142+(5)2=196+25=22114.87单位

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  Example 4
  ::例4

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  Find the distance between (3, 4) and (-1, 3).
  ::查找距离( 3, 4) 和( 1, 3) 之间的距离。

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  Plug in (3, 4) for $\begin{array}{r}(x_1,y_1)\end{array}$ and (-1, 3) for $\begin{array}{r}(x_2,y_2)\end{array}$ and simplify.
  ::插插(3,4),(x1,y1)和(-1,3),(x2,y2)和简化。

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  $$\begin{array}{rl}d& =\sqrt{(-1-3{)}^2+(3-4{)}^2}\\ & =\sqrt{(-4{)}^2+(-1{)}^2}\\ & =\sqrt{16+1}\\ & =\sqrt{17}\approx 4.12units\end{array}$$

  
  ::d=(-1-3)2+(3-4)2=(-4)2+(-1)2+(-1)2+(-1)2=16+1=17=4.12单位

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  Example 5
  ::例5

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  Find the distance between (4, 23) and (8, 14).
  ::查找距离(4、23)和(8、14)之间的距离。

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  Plug in (4, 23) for $\begin{array}{r}(x_1,y_1)\end{array}$ and (8, 14) for $\begin{array}{r}(x_2,y_2)\end{array}$ and simplify.
  ::插插(4,23)至(x1,y1)和(8,14)至(x2,y2)和简化。

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  $$\begin{array}{rl}d& =\sqrt{(8-4{)}^2+(14-23{)}^2}\\ & =\sqrt{(4{)}^2+(-9{)}^2}\\ & =\sqrt{16+81}\\ & =\sqrt{97}\approx 9.85units\end{array}$$

  
  ::d=8-4)2+(14-23)2=(4)2+(-9)2=16+81=97=9.85单位

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  Review
  ::回顾

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  Find the distance between each pair of points. Round your answer to the nearest hundredth.
  ::查找每对点之间的距离。 将您的答复回合到最近的一百个点 。

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  1. (4, 15) and (-2, -1)
     :4,15)和(2,-2,-1)
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  2. (-6, 1) and (9, -11)
     :6,1)和(9,-11)
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  3. (0, 12) and (-3, 8)
     :0,12)和(3,8)
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  4. (-8, 19) and (3, 5)
     :8,19)和(3,5)
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  5. (3, -25) and (-10, -7)
     :3,25)和(10,-7))
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  6. (-1, 2) and (8, -9)
     :-1,2)和(8,9)
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  7. (5, -2) and (1, 3)
     :5, 2, 5, 5, 2 和 1, 3)
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  8. (-30, 6) and (-23, 0)
     :30,6)和(23,0)
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  9. (2, -2) and (2, 5)
     :2,2和2,5)和2,5)
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  10. (-9, -4) and (1, -1)
      :9,4)和(1,1)

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

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  Resources
  ::资源