Course: 5.4 平行线和直直线等量比较

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平行线和直直线等量比较
Comparing Equations of Parallel and Perpendicular Lines
::平行线和直直线等量比较

In this section you will learn how parallel lines and perpendicular lines are related to each other on the coordinate plane . Let’s start by looking at a graph of two parallel lines.
::在本节中,您将了解在坐标平面上的平行线和垂直线是如何相互联系的。让我们先看两条平行线的图表。

We can clearly see that the two lines have different $\begin{aligned} y - \end{aligned}$ intercepts : 6 and –4.
::我们清楚地看到,这两条线有不同的y - intercuts: 6 和 - 4。

How about the slopes of the lines? The of line $\begin{aligned} A \end{aligned}$ is $\begin{aligned} \frac{6 - 2}{0 - (- 2)} = \frac{4}{2} = 2 \end{aligned}$ , and the slope of line $\begin{aligned} B \end{aligned}$ is $\begin{aligned} \frac{0 - (- 4)}{2 - 0} = \frac{4}{2} = 2 \end{aligned}$ . The slopes are the same.
::线的斜坡如何?线A为 6-20-(-2)=42=2,线B为 0-(-4)2-0=42=2。斜坡相同。

Is that significant? Yes. By definition, parallel lines never meet. That means that when one of them slopes up by a certain amount, the other one has to slope up by the same amount so the lines will stay the same distance apart. If you look at the graph above, you can see that for any $\begin{aligned} x - \end{aligned}$ value you pick, the $\begin{aligned} y - \end{aligned}$ values of lines $\begin{aligned} A \end{aligned}$ and $\begin{aligned} B \end{aligned}$ are the same vertical distance apart—which means that both lines go up by the same vertical distance every time they go across by the same horizontal distance. In order to stay parallel, their slopes must stay the same.
::如此重要吗 ? 是的。从定义上看, 平行线从未相遇。这意味着当其中一条向上倾斜一定数量时, 另一条向上倾斜同样数量, 这样线会保持相同的距离。如果您查看上面的图表, 您可以看到对于您所选择的任何 x - 值来说, A 线和 B 线的 Y - 值是相同的垂直距离。这意味着两条线每次通过相同的水平距离时, 都以相同的垂直距离向上上升。为了保持平行, 它们的斜度必须保持不变。

All parallel lines have the same slopes and different $\begin{aligned} y - \end{aligned}$ intercepts.
::所有平行线都有相同的斜坡和不同的y - intercuts。

Now let’s look at a graph of two perpendicular lines.
::现在让我们看看两条垂直线的图。

We can’t really say anything about the $\begin{aligned} y - \end{aligned}$ intercepts. In this example, the $\begin{aligned} y - \end{aligned}$ intercepts are different, but if we moved the lines four units to the right, they would both intercept the $\begin{aligned} y - \end{aligned}$ axis at (0, -2). So perpendicular lines can have the same or different $\begin{aligned} y - \end{aligned}$ intercepts.
::我们无法真正谈论y - 界面。在这个例子中, y - 界面是不同的, 但是如果我们把线4 单位移到右边, 他们就会在 0, - 2 时拦截y - 轴。所以垂直线可以有相同的或不同的 y - 界面。

What about the relationship between the slopes of the two lines?
::那两条线的斜坡之间的关系呢?

To find the slope of line $\begin{aligned} A \end{aligned}$ , we pick two points on the line and draw the blue (upper) right triangle. The legs of the triangle represent the rise and the run . We can see that the slope is $\begin{aligned} \frac{8}{4} \end{aligned}$ , or 2.
::为了找到线A的斜坡, 我们在线上选择两点, 并绘制蓝色( 上方) 右三角形。三角形的腿代表着上升和运行。我们可以看到斜坡是 84 或 2 。

To find the slope of line $\begin{aligned} B \end{aligned}$ , we pick two points on the line and draw the red (lower) right triangle. Notice that the two triangles are identical, only rotated by $\begin{aligned} 90^{\circ} \end{aligned}$ . Where line $\begin{aligned} A \end{aligned}$ goes 8 units up and 4 units right, line $\begin{aligned} B \end{aligned}$ goes 8 units right and 4 units down. Its slope is $\begin{aligned} - \frac{4}{8} \end{aligned}$ , or $\begin{aligned} - \frac{1}{2} \end{aligned}$ .
::为了找到线B的斜坡, 我们在线上选择两个点, 并绘制红( 下) 右三角形。注意这两个三角形是相同的, 仅由 90 来旋转。当线A 向上8 个单位, 向右 4 个单位, 线B 向右 8 个单位, 向下 4 个单位。其斜坡是- 48 或 - 12 。

This is always true for perpendicular lines; where one line goes $\begin{aligned} a \end{aligned}$ units up and $\begin{aligned} b \end{aligned}$ units right, the other line will go $\begin{aligned} a \end{aligned}$ units right and $\begin{aligned} b \end{aligned}$ units down, so the slope of one line will be $\begin{aligned} \frac{a}{b} \end{aligned}$ and the slope of the other line will be $\begin{aligned} - \frac{b}{a} \end{aligned}$ .
::直垂直线总是如此;当一条线向上移动一个单位和b单位,另一条线向右移动一个单位和b单位,所以一条线的斜坡将是 ab,另一条线的斜坡将是-ba。

The slopes of perpendicular lines are always negative reciprocals of each other.
::垂直线的斜坡总是相互负对等的。

Determining When Lines are Parallel or Perpendicular
::确定线线是平行的还是直直的

You can find whether lines are parallel or perpendicular by comparing the slopes of the lines. If you are given points on the lines, you can find their slopes using the formula . If you are given the equations of the lines, re-write each equation in a form that makes it easy to read the slope, such as the slope-intercept form .
::通过比较线条的斜坡,可以发现线条是平行的还是垂直的。如果在线条的斜坡上给定了点,则可以使用公式找到它们的斜坡。如果给定了线条的方程,请将每个方程重写成一种容易读取斜坡的形态,例如斜坡的干扰形态。

1. Determine whether the lines are parallel or perpendicular or neither. One line passes through the points (2, 11) and (-1, 2); another line passes through the points (0, -4) and (-2, -10).
::1. 确定线条是平行线还是垂直线线,或两者兼而有之。一条线通过点(2、11)和(1、2);另一条线通过点(0、4)和(2、10)。

Find the slope of each line and compare them.
::找到每条线的斜坡,比较一下

$\begin{aligned} m_{1} = \frac{2 - 11}{- 1 - 2} = \frac{- 9}{- 3} = 3 and m_{2} = \frac{- 10 - (- 4)}{- 2 - 0} = \frac{- 6}{- 2} = 3 \end{aligned}$
::m1=2-11-1-29-3=3和m2_10-(-4)-2-0_6-2=3

The slopes are equal, so the lines are parallel.
::斜坡是相等的, 线条是平行的。

2. Determine whether the lines are parallel or perpendicular or neither. One line passes through the points (-2, -7) and (1, 5); another line passes through the points (4, 1) and (-8, 4).
::2. 确定线条是平行线还是垂直线线,或两者兼而有之,一条线通过点(-2、7)和点(1、5);另一条线通过点(4、1)和点(8、4)。

$\begin{aligned} m_{1} = \frac{5 - (- 7)}{1 - (- 2)} = \frac{12}{3} = 4 and m_{2} = \frac{4 - 1}{- 8 - 4} = \frac{3}{- 12} = - \frac{1}{4} \end{aligned}$
::m1=5-(-7-7)1-(-2)=123=4和m2=4-8-8-4=3-1214

The slopes are negative reciprocals of each other, so the lines are perpendicular.
::斜坡是相互负对等的,所以线是垂直的。

3. Determine whether the lines are parallel or perpendicular or neither. One line pass es through the points (3, 1) and (-2, -2); another line passes through the points (5, 5) and (4, -6).
::3. 确定线条是平行线还是垂直线线,或两者兼而有之,一条线通过点(3、1和2);另一条线通过点(5、5和4、6),另一条线通过点(5、5和4、6)。

$\begin{aligned} m_{1} = \frac{- 2 - 1}{- 2 - 3} = \frac{- 3}{- 5} = \frac{3}{5} and m_{2} = \frac{- 6 - 5}{4 - 5} = \frac{- 13}{- 1} = 13 \end{aligned}$
::3 - 5=35 和 m2 - 6 - 54 - 5 - 5 - 13 - 1=13

The slopes are not the same or negative reciprocals of each other, so the lines are neither parallel nor perpendicular.
::斜坡互不相同或负对等,因此两条线既不平行,也不垂直。

Examples
::实例

Determine whether the lines are parallel or perpendicular or neither:
::确定线条是平行线还是垂直线线,或两者兼而有之:

Example 1
::例1

$\begin{aligned} 3 x + 4 y = 2 \end{aligned}$ and $\begin{aligned} 8 x - 6 y = 5 \end{aligned}$
::3x+4y=2和8x-6y=5

Write each equation in slope-intercept form:
::以斜坡界面的形式写出每个方程式 :

line 1: $\begin{aligned} 3 x + 4 y = 2 \Rightarrow 4 y = - 3 x + 2 \Rightarrow y = - \frac{3}{4} x + \frac{1}{2} \Rightarrow slope = - \frac{3}{4} \end{aligned}$
::第1行: 3x+4y=24y3x+2434x+12斜度*34

line 2: $\begin{aligned} 8 x - 6 y = 5 \Rightarrow 8 x - 5 = 6 y \Rightarrow y = \frac{8}{6} x - \frac{5}{6} \Rightarrow y = \frac{4}{3} x - \frac{5}{6} \Rightarrow slope = \frac{4}{3} \end{aligned}$
::第2行: 8x- 6y=58x-5=6y_ y=86x-56543x-56斜度=43

The slopes are negative reciprocals of each other, so the lines are perpendicular.
::斜坡是相互负对等的,所以线是垂直的。

Example 2
::例2

$\begin{aligned} 2 x = y - 10 \end{aligned}$ and $\begin{aligned} y = - 2 x + 5 \end{aligned}$
::2x=y- 10 和 y2x+5

line 1: $\begin{aligned} 2 x = y - 10 \Rightarrow y = 2 x + 10 \Rightarrow slope = 2 \end{aligned}$
::第1行: 2x=y- 10_ y=2x+10 斜度=2

line 2: $\begin{aligned} y = - 2 x + 5 \Rightarrow slope = - 2 \end{aligned}$
::行 2: y2x+5 斜度2

The slopes are not the same or negative reciprocals of each other, so the lines are neither parallel nor perpendicular.
::斜坡互不相同或负对等,因此两条线既不平行,也不垂直。

Example 3
::例3

$\begin{aligned} 7 y + 1 = 7 x \end{aligned}$ and $\begin{aligned} x + 5 = y \end{aligned}$
::7y+1=7x和x+5=y

line 1: $\begin{aligned} 7 y + 1 = 7 x \Rightarrow 7 y = 7 x - 1 \Rightarrow y = x - \frac{1}{7} \Rightarrow slope = 1 \end{aligned}$
::第1行:7y+1=7x=7x=7y=7x-1_1}=x-17斜度=1

line 2: $\begin{aligned} x + 5 = y \Rightarrow y = x + 5 \Rightarrow slope = 1 \end{aligned}$
::行 2: x+5=yy=x+5斜度=1

The slopes are the same, so the lines are parallel.
::斜坡是相同的,所以线线是平行的。

Review
::回顾

For 1-10, determine whether the lines are parallel, perpendicular or neither.
::1-10, 确定线线是平行的, 垂直的, 还是两者兼而有之。

One line passes through the points (-1, 4) and (2, 6); another line passes through the points (2, -3) and (8, 1).
::一行通过点(1、2、4和2、6);另一行通过点(2、3和8、1)和点(8、1)。

One line passes through the points (4, -3) and (-8, 0); another line passes through the points (-1, -1) and (-2, 6).
::一行通过点(4,3和8,0);另一行通过点(1,1)和(2,6)。

One line passes through the points (-3, 14) and (1, -2); another line passes through the points (0, -3) and (-2, 5).
::一行通过点(3、14)和点(1、-2);另一行通过点(0、3)和点(2、5)。

One line passes through the points (3, 3) and (-6, -3); another line passes through the points (2, -8) and (-6, 4).
::一行通过点(3,3和6,3);另一行通过点(2,8)和点(6,4)。

One line passes through the points (2, 8) and (6, 0); another line has the equation $\begin{aligned} x - 2 y = 5 \end{aligned}$ .
::一行通过点(2、8)和点(6、0);另一行为方程式x-2y=5。

One line passes through the points (-5, 3) and (2, -1); another line has the equation $\begin{aligned} 2 x + 3 y = 6 \end{aligned}$ .
::一条线通过点(5、3和2,-1);另一条线的方程为2x+3y=6。

Both lines pass through the point (2, 8); one line also passes through (3, 5), and the other line has slope 3.
::两条线通过点(2、8条);一条线也通过点(3、5条),另一条线有斜坡3。

Line 1: $\begin{aligned} 4 y + x = 8 \end{aligned}$ Line 2: $\begin{aligned} 12 y + 3 x = 1 \end{aligned}$
::第1行:4y+x=8 第2行:12y+3x=1

Line 1: $\begin{aligned} 5 y + 3 x = 1 \end{aligned}$ Line 2: $\begin{aligned} 6 y + 10 x = - 3 \end{aligned}$
::第1行:5y+3x=1 第2行:6y+10x=3

Line 1: $\begin{aligned} 2 y - 3 x + 5 = 0 \end{aligned}$ Line 2: $\begin{aligned} y + 6 x = - 3 \end{aligned}$
::第1行:2y-3x+5=0 第2行:y+6x=3

Lines $\begin{aligned} A, B, C, D \end{aligned}$ , and $\begin{aligned} E \end{aligned}$ all pass through the point (3, 6). Line $\begin{aligned} A \end{aligned}$ also passes through (7, 12); line $\begin{aligned} B \end{aligned}$ passes through (8, 4); line $\begin{aligned} C \end{aligned}$ passes through (-1, -3); line $\begin{aligned} D \end{aligned}$ passes through (1, 1); and line $\begin{aligned} E \end{aligned}$ passes through (6, 12).

Are any of these lines perpendicular? If so, which ones? If not, why not?
::这些线中是否有一条是垂直线?如果是,是哪些线?如果没有,为什么不?

Are any of these lines parallel? If so, which ones? If not, why not?
::这些线条是否平行?如果平行,是哪些线条?如果没有平行,为什么不平行?

::A、B、C、D和E线通过点(3、6、6)、A线也通过点(7、12)、B线通过点(8、4)、C线通过点(1-3)、D线通过线(1-3)、D线通过线(1、1);和Epasses线通过线(6、12),这些线中是否有任何一条是垂直的?如果有的话,是哪条线?如果没有,为什么不呢?这些线中是否有一条平行的?如果有,是哪一条平行的?如果有,是哪条线?如果没有,为什么没有?哪些线?如果没有,为什么没有?这些线是平行的呢?如果有,是哪条?如果没有,是哪条线?如果没有,是哪些线?如果没有,是平行的?哪些线呢?如果是平行的?如果是,是哪些线呢?如果是呢?如果是,是哪些呢?如果没有,是哪些线吗?如果没有,为什么?

Review (Answers)
::回顾(答复)

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。

Section outline

General

平行线和直直线等量比较

Comparing Equations of Parallel and Perpendicular Lines ::平行线和直直线等量比较

Determining When Lines are Parallel or Perpendicular ::确定线线是平行的还是直直的

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Comparing Equations of Parallel and Perpendicular Lines
::平行线和直直线等量比较

Determining When Lines are Parallel or Perpendicular
::确定线线是平行的还是直直的

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)