课程： 2.5 查找直直线的等同

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选择节查找直直线的等量

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查找直直线的等量
In most cities, it is very common for the streets to be laid out in a grid format. Below is an example. (Assume the lower left corner is the origin.) Are any of the streets perpendicular? What are the slopes of each street?
::在大多数城市,以网格形式铺设街道是十分常见的。下面就是一个例子。 (假设左下角是源头。 )街道是否与世隔绝?每条街道的斜坡是多少?

Equation of Perpendicular Lines
::直直直线等同

When two lines are perpendicular, they intersect at a $\begin{align*}90^\circ,\end{align*}$ or right, angle. The slopes of two perpendicular lines , are therefore , not the same. Let’s look at the relationship of perpendicular lines in regards to .
::当两条线是垂直的,它们以90角或右角交叉。因此,两条垂直线的斜坡是不一样的。让我们看看垂直线与(......)的关系。

Slopes of Perpendicular Lines
::垂直直线的斜体

Let's work through the following steps to discover the relationship between the slopes of perpendicular lines.
::让我们通过以下步骤来发现垂直线斜坡之间的关系

Tools Needed: Pencil, ruler, protractor, and graph paper
::所需工具:笔纸、标尺、减压器和图表纸

Step 1: Draw a $\begin{align*}x-y\end{align*}$ plane that goes from -5 to 5 in both the $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ directions.
::第1步:在 x 和 y 方向绘制一个从 - 5 到 5 的 x - y 平面。

Step 2: Plot (0, 0) and (1, 3). Connect these to form a line.
::步骤2:绘制(0,0)和(1,3),将这些连接成一条线。

Step 3: Plot (0, 0) and (-3, 1). Connect these to form a second line.
::第3步:绘图(0,0)和(3,1),将这些连接成第二行。

Step 4: Using a protractor, measure the angle formed by the two lines. What is it?
::第4步:使用减速器,测量两条线形成的角度。它是什么?

Step 5: Use slope triangles to find the slope of both lines. What are they?
::第5步:使用斜度三角形找到两条线的斜度。它们是什么?

Step 6: Multiply the slope of the first line times the slope of the second line. What do you get?
::第6步:乘以第一行的斜度乘以第二行的斜度。你得到什么?

We now know that t he lines from Steps 2 and 3 are perpendicular because they form a $\begin{align*}90^\circ\end{align*}$ angle. The slopes are 3 and $\begin{align*}- \frac{1}{3},\end{align*}$ respectively. When multiplied together, the product is -1. This is true of all perpendicular lines.
::我们现在知道,步骤2和步骤3的线条是垂直的,因为它们形成一个 90 角度。斜坡分别为 3 和 13 。乘在一起, 产品是 - 1 。所有垂直线都是如此。

The product of the slopes of two perpendicular lines is -1. If a line has a slope of $\begin{align*}m,\end{align*}$ then the perpendicular slope is $\begin{align*}- \frac{1}{m}\end{align*}$ .
::如果一条线的坡度为米,则垂直坡度为-1米。

Now, let's find the equation of the following perpendicular lines.
::现在,让我们找出以下直径线的方程式。

Find the equation of the line that is perpendicular to $\begin{align*}2x -3y = 15\end{align*}$ and passes through (6, 5).
::查找直径为 2x-3y=15 的直线的方程,然后通过 (6, 5) 。

First, we need to change the line from standard to slope-intercept form .
::首先,我们需要把线从标准线改为斜坡界面。

$\begin{aligned} 2 x - 3 y & = 15 \\ - 3 y & = - 2 x + 15 \\ y & = \frac{2}{3} x - 5 \end{aligned}$
$\begin{align*}2x - 3y &= 15\\ -3y &=-2x + 15\\ y &= \frac{2}{3}x - 5\end{align*}$
::2x-3y=15-3y2x+15y=23x-5

Now, let’s find the perpendicular slope. From the investigation above, we know that the slopes must multiply together to equal -1.
::现在,让我们找到垂直的斜坡。从上面的调查来看,我们知道斜坡必须一起乘以等于-1。

$\begin{aligned} \frac{2}{3} \cdot m & = - 1 \\ \frac{3}{2} \cdot \frac{2}{3} \cdot m & = - 1 \cdot \frac{3}{2} \\ m & = - \frac{3}{2} \end{aligned}$
$\begin{align*}\frac{2}{3} \cdot m &= -1\\ \xcancel{\frac{3}{2} \cdot \frac{2}{3}} \cdot m &= -1 \cdot \frac{3}{2}\\ m &= - \frac{3}{2}\end{align*}$
::23号 23号 31号 32号 32号 32号

Notice that the perpendicular slope is the opposite sign and reciprocals with the original slope. Now, we need to use the given point to find the $\begin{align*}y-\end{align*}$ intercept .
::注意垂直斜坡与原来的斜坡是相反的标志和对等标志。现在,我们需要使用给定的点来找到y- intercut 。

$\begin{aligned} 5 & = - \frac{3}{2} (6) + b \\ 5 & = - 9 + b \\ 14 & = b \end{aligned}$
$\begin{align*}5 &= - \frac{3}{2}(6) + b\\ 5 &= -9 + b\\ 14 &= b\end{align*}$
::532(6)+b59+b14=b

The equation of the line that is perpendicular to $\begin{align*}y = \frac{2}{3}x - 5 \end{align*}$ is $\begin{align*}y = - \frac{3}{2}x + 14\end{align*}$ .
::与 y= 23x- 5 垂直的直线的方程式是 y 32x+ 14。

If we write these lines in standard form , the equations would be $\begin{align*}2x - 3y = 15\end{align*}$ and $\begin{align*}3x + 2y = 28,\end{align*}$ respectively.
::如果我们以标准格式写下这些线条, 方程将分别为 2x-3y=15 和 3x+2y=28。

Write the equation of the line that passes through (4, -7) and is perpendicular to $\begin{align*}y = 2\end{align*}$ .
::写入通过(4,7)的直线的方程(4,7),该直线与y=2垂直。

The line $\begin{align*}y = 2\end{align*}$ does not have a $\begin{align*}x-\end{align*}$ term , meaning it has no slope and a horizontal line. Therefore, to find the perpendicular line that passes through (4, -7), it would have to be a vertical line. Only need the $\begin{align*}x-\end{align*}$ coordinate. The perpendicular line would be $\begin{align*}x = 4\end{align*}$ .
::y=2 线没有 x- term, 意思是它没有斜度和水平线。因此, 要找到通过( 4, - 7) 的垂直线, 它必须是一条垂直线。只要需要 x - 坐标。垂直线将是 x= 4 。

Write the equation of the line that passes through (6, -10) and is perpendicular to the line that passes through (4, -6) and (3, -4).
::写出通过(6,10)的线的方程式,该线的方程式与通过(4,6)和(3,4)的线的方程式是垂直的。

First, we need to find the slope of the line that our line will be perpendicular to. Use the points (4, -6) and (3, -4) to find the slope.
::首先,我们需要找到一条线的斜坡,这条线将直线与这条线相连接。用点数(4,6)和点数(3,4)来找到斜坡。

$\begin{aligned} m = \frac{- 4 - (- 6)}{3 - 4} = \frac{2}{- 1} = - 2 \end{aligned}$
$\begin{align*}m = \frac{-4-(-6)}{3-4} = \frac{2}{-1} = -2\end{align*}$
::m*4 - (-6)3 - 4=2 - 1 *2

Therefore, the perpendicular slope is the opposite sign and the reciprocal of -2. That makes the slope $\begin{align*}\frac{1}{2}\end{align*}$ . Use the point (6, -10) to find the $\begin{align*}y-\end{align*}$ intercept.
::因此,垂直斜坡是相反的标志,是-2.的对等标志,使斜坡成为12. 利用点(6,10)找到y-intercut。

$\begin{aligned} - 10 & = \frac{1}{2} (6) + b \\ - 10 & = 3 + b \\ - 7 & = b \end{aligned}$
$\begin{align*}-10 &= \frac{1}{2}(6) + b\\ -10 &= 3 + b \\ -7 &= b\end{align*}$
::- 10=12(6)+b-10=3+b-7=b

The equation of the perpendicular line is $\begin{align*}y = \frac{1}{2}x - 7\end{align*}$ .
::垂直直线的方程式是y=12x-7。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find out if any of the streets in the photo are perpendicular by determining the slope of each street.
::早些时候,有人要求你通过确定每一条街道的斜坡,了解照片中的街道是否与众不同。

Using the graphed endpoints of each street, we find the slope of A St. is $\begin{align*}- \frac{1}{2}\end{align*}$ , 7th Ave. is $\begin{align*}2\end{align*}$ , and the slope is Lincoln Blvd. is $\begin{align*}\frac{1}{2}\end{align*}$ . Therefore, because they are negative reciprocals of each other, A St. and 7th Ave. are perpendicular. Lincoln Blvd. is neither perpendicular nor parallel to 7th Ave. or A Street.
::使用每条街道的图形端点,我们发现A街的斜坡是-12, 7Ave是-2, 而斜坡是林肯Blvd。是12。因此,A街和7Aave是负面对等的,因此,A街和A街是垂直的。Lincoln Blvd.既不是垂直的,也不是与7Aave或A街平行的。

Example 2
::例2

Find the equation of the line that is perpendicular to $\begin{align*}x - 2y = 8\end{align*}$ and passes through (4, -3).
::查找直径为 x-2y=8 的直线的方程式,然后通过(4, 3) 。

First, we need to change this line from standard form to slope-intercept form.
::首先,我们需要将这条线从标准格式改为斜坡拦截形式。

$\begin{aligned} x - 2 y & = 8 \\ - 2 y & = - x + 8 \\ y & = \frac{1}{2} x - 4 \end{aligned}$
$\begin{align*}x -2y &= 8\\ -2y &= -x + 8\\ y &=\frac{1}{2}x - 4\end{align*}$
::x-2y=8--2yx+8y=12x-4

The perpendicular slope will be -2. Let's find the new $\begin{align*}y-\end{align*}$ intercept.
::垂直斜坡将是 -2. 让我们找到新的 y - interview 。

$\begin{aligned} - 3 & = - 2 (4) + b \\ - 3 & = - 8 + b \\ 5 & = b \end{aligned}$
$\begin{align*}-3 &= -2(4) + b\\ -3 &= -8 + b \\ 5 &= b\end{align*}$
::- 32(4)+b-38+b5=b

The equation of the perpendicular line is $\begin{align*}y = -2x + 5\end{align*}$ or $\begin{align*}2x + y = 5\end{align*}$ .
::垂直直线的方程式是 y2x+5 或 2x+y=5 。

Example 3
::例3

Find the equation of the line that passes through (-8, 7) and is perpendicular to the line that passes through (6, -1) and (1, 3).
::查找通过(8, 7)的线的方程,该线与通过(6, 1)和(1, 3)的线垂直。

First, find the slope between (6, -1) and (1, 3).
::首先,找到在(6,-1)和(1,3)之间的斜坡。

$\begin{aligned} m = \frac{- 1 - 3}{6 - 1} = \frac{- 4}{5} = - \frac{4}{5} \end{aligned}$
$\begin{align*}m = \frac{-1-3}{6-1} = \frac{-4}{5} = -\frac{4}{5}\end{align*}$
::m1-36 -1 4545

From this, the slope of the perpendicular line will be $\begin{align*}\frac{5}{4}\end{align*}$ . Now, use (-8, 7) to find the $\begin{align*}y-\end{align*}$ intercept.
::从这里,垂直直线的斜坡将是54。现在,使用( 8, 7) 来查找 y- intercut 。

$\begin{aligned} 7 & = \frac{5}{4} (- 8) + b \\ 7 & = - 10 + b \\ 17 & = b \end{aligned}$
$\begin{align*}7 &= \frac{5}{4}(-8) + b\\ 7 &= -10 + b \\ 17 &= b\end{align*}$
::7=54(-8)+b7=10+b17=b

The equation of the perpendicular line is $\begin{align*}y = \frac{5}{4}x + 17\end{align*}$ .
::垂直直线的方程式是y=54x+17。

Example 4
::例4

Are $\begin{align*}x - 4y = 8\end{align*}$ and $\begin{align*}2x + 8y = -32\end{align*}$ parallel, perpendicular or neither?
::x-4y=8和2x+8y32是平行的、垂直的还是两者兼有?

To determine if the two lines are parallel or perpendicular, we need to change them both into slope-intercept form.
::为了确定这两条线是平行的还是垂直的,我们需要将这两条线都改变为斜坡间接形式。

$\begin{aligned} x - 4 y & = 8 2 x + 8 y = - 32 \\ - 4 y & = - x + 8 a n d 8 y = - 2 x - 32 \\ y & = \frac{1}{4} x - 2 y = - \frac{1}{4} x - 4 \end{aligned}$
$\begin{align*}x - 4y &= 8 \qquad \qquad \qquad 2x + 8y = -32\\ -4y &= -x + 8 \quad and \qquad \quad \ 8y = -2x -32\\ y &=\frac{1}{4}x - 2 \qquad \qquad \qquad \ y = - \frac{1}{4}x - 4\end{align*}$
::x - 4y= 82x+8y_ 32- 4y%x+8和 8y2x-32y=14x-2y14x- 4

Now, just look at the slopes. One is $\begin{align*}\frac{1}{4}\end{align*}$ and the other is $\begin{align*}- \frac{1}{4}\end{align*}$ . They are not the same, so they are not parallel. To be perpendicular, the slopes need to be reciprocals, which they are not. Therefore, these two lines are not parallel or perpendicular.
::现在,请看看斜坡。一个是14,另一个是14。它们不一样,所以不是平行的。要垂直,斜坡必须是对等的,而不是对等的。因此,这两条线不是平行的,也不是对等的。

Review
::回顾

Find the equation of the line, given the following information. You may leave your answer in slope-intercept form.
::查找线条的方程式, 给出以下信息。您可以将答案保留在斜坡界面中。

Passes through (4, 7) and is perpendicular to $\begin{align*}x - y = -5\end{align*}$ .
::通过(4、7)和直通X-y5的通道。

Passes through (-6, -2) and is perpendicular to $\begin{align*}y = 4\end{align*}$ .
::通过(-6,-2),与y=4垂直。

Passes through (4, 5) and is perpendicular to $\begin{align*}y = - \frac{1}{3}x - 1\end{align*}$ .
::通过(4,5),与y13x-1垂直。

Passes through (1, -9) and is perpendicular to $\begin{align*}x = 8\end{align*}$ .
::通过(1,-9),直通x=8。

Passes through (0, 6) and perpendicular to $\begin{align*}x - 4y = 10\end{align*}$ .
::通过(0、6)和直通X-4y=10的通过。

Passes through (-12, 4) and is perpendicular to $\begin{align*}y = -3x + 5\end{align*}$ .
::通过( 12, 4) 并且与 y3x+5 垂直。

Passes through the $\begin{align*}x-\end{align*}$ intercept of $\begin{align*}2x - 3y = 6\end{align*}$ and perpendicular to $\begin{align*}x + 6y = -3\end{align*}$ .
::通过x-3y=6和直通 x+6y=3的x- intercept 和x+6y=6的X- intercept。

Passes through (7, -8) and is perpendicular to $\begin{align*}2x + 5y = 14\end{align*}$ .
::通过(7,-8),垂直为2x+5y=14。

Passes through (1, 3) and is perpendicular to the line that passes through (-6, 2) and (-4, 6).
::通过(1,3)和(6,2)和(4,6)的通道,与通过(6,2)和(4,6)的线路相连。

Passes through (3, -10) and is perpendicular to the line that passes through (-2, 2) and (-8, 1).
::通过(3, - 10)的通行证,与通过(-2,2)和(-8,1)的路线相对应。

Passes through (-4, -1) and is perpendicular to the line that passes through (-15, 7) and (-3, 3).
::通过(4-4-1),与通过(15、7)和(3、3)。

Are the pairs of lines parallel, perpendicular or neither?
::这对线是平行的, 垂直的还是两者的?

$\begin{align*}4x + 2y = 5\end{align*}$ and $\begin{align*}5x - 10y = -20\end{align*}$
::4x+2y=5和5x-10y20

$\begin{align*}9x + 12y = 8\end{align*}$ and $\begin{align*}6x + 8y = -1\end{align*}$
::9x+12y=8和6x+8y=1

$\begin{align*}5x - 5y = 20\end{align*}$ and $\begin{align*}x + y = 7\end{align*}$
::5x-5y=20和x+y=7

$\begin{align*}8x -4y = 12\end{align*}$ and $\begin{align*}4x - y = -15\end{align*}$
::8-4y=12和4x-y15

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。

章节大纲

常规

查找直直线的等量

Equation of Perpendicular Lines ::直直直线等同

Slopes of Perpendicular Lines ::垂直直线的斜体

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Equation of Perpendicular Lines
::直直直线等同

Slopes of Perpendicular Lines
::垂直直线的斜体

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)