11.12 中点公式

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  中点公式

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  The Midpoint Formula 
  ::中点公式

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  Here  you will learn how to find the point exactly half way between two points.
  ::在这里您将学习如何在两点之间 找到正中一半的点。

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  Find the coordinates of the point that is in the middle of the line segment connecting the points $\begin{align*}A = (-7, -2)\end{align*}$ and $\begin{align*}B = (3, -8)\end{align*}$ .
  ::查找连接A=(-7)-2点和B=(3)-8点的线段中间点的坐标。

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  Let’s start by graphing the two points:
  ::让我们先用图表绘制两点:

  

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  We see that to get from point $\begin{align*}A\end{align*}$ to point $\begin{align*}B\end{align*}$ we move 6 units down and 10 units to the right.
  ::我们看到从A点到B点 我们把6个单位往下移动 10个单位往右移动

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  In order to get to the point that is halfway between the two points, it makes sense that we should move half the vertical distance and half the horizontal distance—that is, 3 units down and 5 units to the right from point $\begin{align*}A\end{align*}$ .
  ::为了达到两点之间的中间点,我们应该将垂直距离的一半和水平距离的一半,即从A点向下3个单位和5个单位向右移动,这是有道理的。

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  The midpoint is $\begin{align*}M = (-7 +5, -2 - 3) = (-2, -5)\end{align*}$ .
  ::中点为M=(-7+5,-2-3)=(-2,-5)。

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  The Midpoint Formula
  ::中点公式

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  We now want to generalize this method in order to find a formula for the midpoint of a line segment.
  ::我们现在要普遍采用这种方法,以便找到一条线段中点的公式。

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  Let’s take two general points $\begin{align*}A = (x_1, y_1)\end{align*}$ and $\begin{align*}B = (x_2, y_2)\end{align*}$ and mark them on the coordinate plane :
  ::让我们以 A = (x1,y1) 和 B = (x2,y2) 作两个一般点并在坐标平面上标记 :

  

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  We see that to get from $\begin{align*}A\end{align*}$ to $\begin{align*}B\end{align*}$ , we move $\begin{align*}x_2 - x_1\end{align*}$ units to the right and $\begin{align*}y_2 - y_1\end{align*}$ units up.
  ::我们可以看到,从A到B, 我们移动 x2 - x1 单位到右边 和 y2 -y1 单位向上。

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  In order to get to the half-way point, we need to move $\begin{align*}\frac{x_2-x_1}{2}\end{align*}$ units to the right and $\begin{align*}\frac{y_2-y_1}{2}\end{align*}$ up from point $\begin{align*}A\end{align*}$ . Thus the midpoint $\begin{align*}M\end{align*}$ is at $\begin{align*}\left(x_1+\frac{x_2-x_1}{2}, y_1+\frac{y_2-y_1}{2}\right)\end{align*}$ .
  ::为了达到中途点,我们需要从A点向右移动 x2 - x12 单位和 y2 -y12 单位。 因此中点 M 值为 (x1+x2 - x12,y1+y2 -y12) 。

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  This simplifies to $\begin{align*}M =\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\end{align*}$ . This is the Midpoint Formula:
  ::此简化为 M= (x1+x22,y1+y22) 。 这是中点公式 :

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  The midpoint of the line segment connecting the points $\begin{align*}(x_1, y_1)\end{align*}$ and $\begin{align*}(x_2, y_2)\end{align*}$ is $\begin{align*}\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\end{align*}$ .
  ::连接点(x1,y1)和点(x2,y2)的线段中点是(x1+x22,y1+y22)。

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  It should hopefully make sense that the midpoint of a line is found by taking the average values of the $\begin{align*}x\end{align*}$ and $\begin{align*}y-\end{align*}$ values of the endpoints.
  ::希望用终点的 x 和 y - 值的平均值来找到线的中点是有道理的。

   

   

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  Finding the Midpoint 
  ::寻找中点

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  Find the midpoint between the following points.
  ::在以下各点之间查找中点。

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  Let’s apply the Midpoint Formula: $\begin{align*}\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\end{align*}$
  ::让我们应用中点公式x1+x22,y1+y22)

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  a) (-10,2)
  ::a) (10,2)

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  the midpoint of (-10, 2) and (3, 5) is $\begin{align*}\left(\frac{-10+3}{2}, \frac{2+5}{2}\right)=\left(\frac{-7}{2}, \frac{7}{2}\right)=\underline{\underline{(-3.5,3.5)}}\end{align*}$
  ::中点(-10,2)和中点(3,5)为(-10+32,2+52)=(-72,72)=(-3.5,3.5)__

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  b) (3,5)
  ::b) (3,5)

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  the midpoint of (3, 6) and (7, 6) is $\begin{align*}\left(\frac{3+7}{2}, \frac{6+6}{2}\right)=\left(\frac{10}{2}, \frac{12}{2}\right)=\underline{\underline{(5,6)}}\end{align*}$
  ::中点(3,6)和中点(7,6)为(3+72,6+62)=(102,122)=(5,6)__

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  Finding  Endpoints
  ::寻找终点

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  A line segment whose midpoint is (2, -6) has an endpoint of (9, -2). What is the other endpoint?
  ::中点为(2,6)的线段端点为(9,2)的线段端点为(9,2),另一端点是什么?

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  In this problem we know the midpoint and we are looking for the missing endpoint.
  ::在这个问题上,我们知道中点,我们正在寻找缺失的终点。

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  The midpoint is (2, -6).
  ::中点为2,6。

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  One endpoint is $\begin{align*}(x_1, x_2) = (9, -2)\end{align*}$ .
  ::一个终点是(x1,x2)=(9,-2)。

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  Let’s call the missing point $\begin{align*}(x, y)\end{align*}$ .
  ::让我们把缺点(x,y)称为(x,y) 。

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  We know that the $\begin{align*}x-\end{align*}$ coordinate of the midpoint is 2, so: $\begin{align*}2=\frac{9+x_2}{2} \Rightarrow 4=9+x_2 \Rightarrow x_2=-5\end{align*}$
  ::我们知道中点的x- 坐标是 2, 所以: 2= 9+x22+4= 9+x2+x2+x2+x2=5

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  We know that the $\begin{align*}y-\end{align*}$ coordinate of the midpoint is -6, so:
  ::我们知道中点的Y坐标是 -6,所以:

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  $$\begin{align*}-6=\frac{-2+y_2}{2} \Rightarrow -12=-2+y_2 \Rightarrow y_2=-10\end{align*}$$
  ::-=YTET -伊甸园字幕组=- 翻译:

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  The missing endpoint is (-5, -10).
  ::缺失的终点是 (5 - 10) 。

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  Here’s another way to look at this problem: To get from the endpoint (9, -2) to the midpoint (2, -6), we had to go 7 units left and 4 units down. To get from the midpoint to the other endpoint, then, we would need to go 7 more units left and 4 more units down, which takes us to (-5, -10).
  ::这个问题的另一种观点是:从终点(9,-2)到中点(2,6),我们不得不左转7个单位,向下走4个单位。 从中点到另一端点,我们需要再左转7个单位,向下走4个单位,这把我们带到(5,10),再往下走7个单位,再往下走4个单位。

   

   

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  Example
  ::示例示例示例示例

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  Example 1
  ::例1

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  Find the midpoint between the points (4, -5) and (-4, 5).
  ::查找点(4,5)和点(4,5)之间的中点。

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  Let’s apply the Midpoint Formula: $\begin{align*}\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\end{align*}$
  ::让我们应用中点公式x1+x22,y1+y22)

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  The midpoint of (4, -5) and (-4, 5) is $\begin{align*}\left(\frac{4-4}{2}, \frac{-5+5}{2}\right)=\left(\frac{0}{2}, \frac{0}{2}\right)=\underline{\underline{(0,0)}}\end{align*}$
  ::中点(4,5)和中点(4,5)为(4-42,-5+52)=(02,02)=(0,0)__

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  Review 
  ::回顾

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  Find the midpoint of the line segment joining the two points.
  ::查找连接两个点的线段的中点。

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  1. (3, -4) and (6, 1)
     :3,4)和(6,1)
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  2. (2, -3) and (2, 4)
     :2,3和2,4)
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  3. (4, -5) and (8, 2)
     :4、5和5)和(8、2)
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  4. (1.8, -3.4) and (-0.4, 1.4)
     :1.8,3.4)和(-0.4,1.4)
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  5. (5, -1) and (-4, 0)
     :5,-1)和(4,0)
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  6. (10, 2) and (2, -4)
     :10、2和2,4)
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  7. (3, -3) and (2, 5)
     :3,3,3)和(2,5)
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  8. An endpoint of a line segment is (4, 5) and the midpoint of the line segment is (3, -2). Find the other endpoint.
     ::线段的终点是(4,5),线段的中点是(3,2),找到另一个终点。
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  9. An endpoint of a line segment is (-10, -2) and the midpoint of the line segment is (0, 4). Find the other endpoint.
     ::线段的终点是(-10,-2),线段的中点是(0, 4)。
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  10. Find a point that is the same distance from (4, 5) as it is from (-2, -1), but is not the midpoint of the line segment connecting them.
      ::查找一个与( 2 - 1) 相同的距离( 4 、 5) 的点, 但不是连接它们的线段的中点 。

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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 。