3.9 坐标平面上的直直线

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  坐标平面上的直直线

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  Here you'll learn that the slopes of perpendicular lines are negative reciprocals of each other. You'll then apply this fact to determine if two lines are perpendicular and to find what their equations are.
  ::这里您会知道垂直线的斜坡是相互对等的负值。 然后您会运用这个事实来确定两条线是否垂直, 并找到它们的方程。

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  Perpendicular Lines in the Coordinate Plane
  ::坐标平面上的直直线

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  Perpendicular lines are two lines that intersect at a 90 degree, or right, angle . In the coordinate plane , that would look like this:
  ::垂直线是两条线,在90度或右边角交叉。在坐标平面上,这看起来像:

  

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  If we take a closer look at these two lines, the of one is -4 and the other is 1/4.
  ::如果我们仔细看一看这两条线, 一条是 -4,另一条是1/4。

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  This can be generalized to any pair of perpendicular lines in the coordinate plane. The slopes of perpendicular lines are opposite reciprocals of each other.
  ::这可以推广到坐标平面上任何一条垂直线,垂直线的斜坡是对等的。

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  What if you were given two perpendicular lines in the coordinate plane? What could you say about their slopes? 
  ::如果你在座标飞机上有两条直直线呢?

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  Calculating the Slope of Perpendicular Lines
  ::计算垂直线的曲线

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  Find the slope of the perpendicular lines to the lines below.
  ::将垂直直线的斜坡定位到下面的直线上。

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  a) $\begin{align*}y=2x+3\end{align*}$
  ::a) y=2x+3

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  $\begin{align*}m = 2\end{align*}$ , so  $\begin{align*}m_\perp\end{align*}$  is the reciprocal and negative,  $\begin{align*}m_\perp=-\frac{1}{2}\end{align*}$ .
  ::m=2,所以m是对应的和负的,m12。

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  b) $\begin{align*}y=-\frac{2}{3}x-5\end{align*}$
  :b) y 23x-5

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  $\begin{align*}m=-\frac{2}{3}\end{align*}$ , take the reciprocal and make the slope positive,  $\begin{align*}m_\perp=\frac{3}{2}\end{align*}$
  ::23米23乘以对等,使斜坡正,32米32

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  c) $\begin{align*}y=x+2\end{align*}$
  ::c) y=x+2

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  Because there is no number in front of $\begin{align*}x\end{align*}$ , the slope is 1. The reciprocal of 1 is 1, so the only thing to do is make it negative, $\begin{align*}m_\perp=-1\end{align*}$ .
  ::因为x前面没有数字, 斜坡是 1 。 1 的对等值是 1, 所以唯一要做的就是让它负, m1 。

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  Finding the Equation of a Perpendicular Line 
  ::查找直直线的等量

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  Find the equation of the line that is perpendicular to $\begin{align*}y=-\frac{1}{3}x+4\end{align*}$ and passes through (9, -5).
  ::查找与 y13x+4 垂直的直线的方程,然后通过 (9,5) 。

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  First, the slope is the reciprocal and opposite sign of $\begin{align*}-\frac{1}{3}\end{align*}$ . So, $\begin{align*}m = 3\end{align*}$ . Now, we need to find the $\begin{align*}y-\end{align*}$ intercept. 4 is the $\begin{align*}y-\end{align*}$ intercept of the given line, not our new line . We need to plug in 9 for $\begin{align*}x\end{align*}$ and -5 for $\begin{align*}y\end{align*}$ to solve for the new $\begin{align*}y-\end{align*}$ intercept $\begin{align*}(b)\end{align*}$ .
  ::首先,斜坡是-13的对等和相反的符号。 所以, m=3。 现在我们需要找到y- interfict。 4 是给定线的y- intervictive。 4 是给定线的y- intervictive, 而不是我们的新线 。 我们需要在 x 和 - 5 中插入 9 , y 才能解决新- intervictive (b) 。

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  $$\begin{align*}-5 & = 3(9)+b\\ -5 & = 27 + b \qquad \text{Therefore, the equation of line is} \ y=3x-32.\\ -32 & = b\end{align*}$$
  ::-5=3(9)+b=5=27+b因此,线的方程式是y=3x-32.-32.-32=b

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  Graphing the Equation of a Line 
  ::绘制线条的方形

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  Graph $\begin{align*}3x-4y=8\end{align*}$ and $\begin{align*}4x+3y=15\end{align*}$ . Determine if they are perpendicular.
  ::图3 - 4y=8和4x+3y=15。确定它们是垂直的。

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  First, we have to change each equation into slope-intercept form. In other words, we need to solve each equation for $\begin{align*}y\end{align*}$ .
  ::首先,我们必须将每个方程式改变为斜坡界面。换句话说,我们需要为y 解决每个方程式。

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  $$\begin{align*}3x-4y & = 8 && 4x+3y = 15\\ -4y & = -3x+8 && 3y = -4x + 15\\ y & = \frac{3}{4}x-2 && y = -\frac{4}{3}x+5\end{align*}$$
  ::3x-4y=84x+3y=15-4y3x+83y4x+15y=34x-2y43x+5

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  Now that the lines are in slope-intercept form (also called $\begin{align*}y-\end{align*}$ intercept form), we can tell they are perpendicular because their slopes are opposite reciprocals.
  ::现在线以斜坡界面的形式(也称为 y - interphysict form)出现, 我们可以看出来它们是垂直的, 因为它们的斜坡是相反的对等的 。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Determine which of the following pairs of lines are perpendicular.
  ::确定以下几条线的哪一对是垂直线。

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  • $\begin{align*}y=-2x+3\end{align*}$ and $\begin{align*}y=\frac{1}{2}x+3\end{align*}$
    ::y2x+3和y=12x+3

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  • $\begin{align*}y=4x-2\end{align*}$ and $\begin{align*}y=4x+5\end{align*}$
    ::y=4x-2和y=4x+5 y=4x-2和y=4x+5

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  • $\begin{align*}y=-x+5\end{align*}$ and $\begin{align*}y=x+1\end{align*}$
    ::yx+5和y=x+1 yx+5和y=x+1

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  Two lines are perpendicular if their slopes are opposite reciprocals. The only pairs of lines this is true for is the first pair, because $\begin{align*}-2\end{align*}$ and $\begin{align*}\frac{1}{2}\end{align*}$ are opposites and reciprocals.
  ::如果两条线的斜坡是对等的,则两条线是垂直的。 唯一的一对线是第一对,因为-2和12是对等的和对等的。

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  Example 2
  ::例2

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  Find the equation of the line that is perpendicular to the line $\begin{align*}y=2x+7\end{align*}$ and goes through the point (2, -2).
  ::查找与 y=2x+7 线垂直的直线的方程式,然后通过点(2, - 2) 。

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  The perpendicular line goes through (2, -2), but the slope is $\begin{align*}-\frac{1}{2}\end{align*}$ because we need to take the opposite reciprocal of $\begin{align*}2\end{align*}$ .
  ::垂直线通过(2,2),但斜坡是-12,因为我们需要对等2。

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  $$\begin{align*}y & = -\frac{1}{2}x+b\\ -2 & = -\frac{1}{2}(2) + b\\ -2 & = -1+b\\ -1 & =b\end{align*}$$
  ::y12x+b-212(2)+b-21+b-1=b

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  The equation is $\begin{align*}y = -\frac{1}{2}x-1\end{align*}$ .
  ::方程式是y12x-1。

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  Example 3
  ::例3

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  Give an example of a line that is perpendicular to the line $\begin{align*}y=\frac{2}{3}x-4\end{align*}$ .
  ::举例来说,直线与直线y=23x-4垂直。

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  3. Any line perpendicular to $\begin{align*}y=\frac{2}{3}x-4\end{align*}$ will have a slope of $\begin{align*}-\frac{3}{2}\end{align*}$ . Any equation of the form $\begin{align*}y=-\frac{3}{2}x+b\end{align*}$ will work. 
  ::3. 与y=23x-4垂直的任何直线的斜度为-32。

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  Review 
  ::回顾

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  1. Determine which of the following pairs of lines are perpendicular.
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     1. $\begin{align*}y=-3x+1\end{align*}$ and $\begin{align*}y=3x-1\end{align*}$
        ::y3x+1和y=3x-1
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     2. $\begin{align*}2x-3y=6\end{align*}$ and $\begin{align*}3x+2y=6\end{align*}$
        ::2x-3yy=6和3x+2y=6
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     3. $\begin{align*}5x+2y=-4\end{align*}$ and $\begin{align*}5x+2y=8\end{align*}$
        ::5x+2y4和5x+2y=8
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     4. $\begin{align*}x-3y=-3\end{align*}$ and $\begin{align*}x+3y=9\end{align*}$
        ::x-3y3 和 x+3y=9
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     5. $\begin{align*}x+y=6\end{align*}$ and $\begin{align*}4x+4y=-16\end{align*}$
        ::x+y=6和4x+4y16
     
     ::确定以下几对直线是垂直的。 y3x+1 和 y= 3x-1 2x-3y=6 和 3x+2y=6 5x+2y4 和 5x+2y=8 x- 3y3 和 x+3y=9 x+y=6 和 4x+4y}16

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  Determine the equation of the line that is perpendicular to the given line, through the given point.
  ::确定通过给定点与给定线垂直的线条的方程。

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  2. $\begin{align*}y=x-1; \ (-6, \ 2)\end{align*}$
     ::y=x-1; (-6, 2)
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  3. $\begin{align*}y=3x+4; \ (9, \ -7)\end{align*}$
     ::y=3x+4; (9, -7)
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  4. $\begin{align*}5x-2y=6; \ (5, \ 5)\end{align*}$
     ::5x-2yy=6; (5,5)
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  5. $\begin{align*}y = 4; \ (-1, \ 3)\end{align*}$
     ::y=4; (-1, 3)
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  6. $\begin{align*}x = -3; \ (1, \ 8)\end{align*}$
     ::x% 3; (1, 8)
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  7. $\begin{align*}x - 3y = 11; \ (0, \ 13)\end{align*}$
     ::x3y=11; (0, 13)

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  Determine if each pair of lines is perpendicular or not.
  ::确定每条线的直线是否垂直。

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  For the line and point below, find a perpendicular line, through the given point.
  ::对于下方的线和点,通过给定点找到垂直线。

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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 。