Course: 1.2 两点之间的距离

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section 两点之间的距离

Collapse Expand
两点之间的距离
Measuring Distance Between Two Points
::测量两点之间的距离

Distance is the measure of length between two points. To measure is to determine how far apart two geometric objects are. The most common way to measure distance is with a ruler. Inch-rulers are usually divided up by eighth-inch (or 0.125 in) segments. Centimeter rulers are divided up by tenth-centimeter (or 0.1 cm) segments. Note that the distance between two points is the absolute value of the difference between the numbers shown on the ruler. This implies that you do not need to start measuring at “0”, as long as you subtract the first number from the second.
::两点之间的距离是测量两个几何天体之间的距离。测量距离的最常用方法是用标尺测量距离。英寸标尺通常用八英寸( 或0. 125英寸) 的区块分隔开来。厘米标尺用十厘米( 或0. 1厘米) 的区块分隔开来。请注意, 两点之间的距离是标尺上显示的数字之间的差数的绝对值。这意味着只要从第二个区块中减去第一个数, 您不需要开始测量“ 0 ” 。

The segment addition postulate states that if $\begin{aligned} A \end{aligned}$ , $\begin{aligned} B \end{aligned}$ , and $\begin{aligned} C \end{aligned}$ are collinear and $\begin{aligned} B \end{aligned}$ is between $\begin{aligned} A \end{aligned}$ and $\begin{aligned} C \end{aligned}$ , then $\begin{aligned} A B + B C = A C \end{aligned}$ .
::增加部分的假设是,如果A、B和C为圆线性,B介于A和C之间,则AB+BC=AC。

You can find the distances between points in the $\begin{aligned} x \end{aligned}$ – $\begin{aligned} y \end{aligned}$ plane if the lines are horizontal or vertical. If the line is vertical, find the change in the $\begin{aligned} y - \end{aligned}$ coordinates. If the line is horizontal, find the change in the $\begin{aligned} x - \end{aligned}$ coordinates.
::如果线条是水平或垂直的,您可以在 x - y 平面中找到点之间的距离。如果线条是垂直的,请在 y - 坐标处找到变化。如果线条是水平的,请在 x - 坐标处找到变化。

Suppose you want to measure your height , but the measuring tape you have is old and the end is broken off. If the tape now starts at 6 cm and reads 138 cm from the floor to the top of your head, how tall are you?
::假设你想测量你的身高,但你的测量磁带是旧的,末端是折断的。如果磁带现在从6厘米开始,读到从地板到头顶的138厘米,那么你有多高?

Examples
::实例

Example 1
::例1

What is the distance marked on the ruler below? The ruler is in centimeters.
::标尺下面的距离是多少? 标尺是厘米。

Subtract one endpoint from the other. The line segment spans from 3 cm to 8 cm. $\begin{aligned} | 8 - 3 | = | 5 | = 5 \end{aligned}$
::从另一端减去一个端点。线段的长度从 3 厘米到 8 厘米不等。 @ @ 8- 3 @ 5 @ 5 @ 5 @ label

The line segment is 5 cm long. Notice that you also could have done $\begin{aligned} | 3 - 8 | = | - 5 | = 5 \end{aligned}$ .
::线段长度为 5 厘米。请注意, 您也可以做 3 - 8 5 5 。

Example 2
::例2

Make a sketch of $\begin{aligned} \bar{O P} \end{aligned}$ , where $\begin{aligned} Q \end{aligned}$ is between $\begin{aligned} O \end{aligned}$ and $\begin{aligned} P \end{aligned}$ .
::给O和P之间的Q线画个草图

Draw $\begin{aligned} \bar{O P} \end{aligned}$ first, then place $\begin{aligned} Q \end{aligned}$ on the segment.
::先绘制 OP, 然后在段上放置 Q 。

Example 3
::例3

What is the distance between the two points shown below?
::下面列出的两点之间的距离是多少?

Because this line is vertical, look at the change in the $\begin{aligned} y - \end{aligned}$ coordinates.
::因为这条线是垂直的, 看看Y座标的变化。

$\begin{aligned} | 9 - 3 | = | 6 | = 6 \end{aligned}$

The distance between the two points is 6 units.
::两点之间的距离是6个单位。

Example 4
::例4

In the picture from Example 2, if $\begin{aligned} O P = 17 \end{aligned}$ and $\begin{aligned} Q P = 6 \end{aligned}$ , what is $\begin{aligned} O Q \end{aligned}$ ?
::从例2可以看出,如果OP=17和QP=6,什么是OQ?

Use the Segment Addition Postulate.
::使用线段附加假设。

$\begin{aligned} O Q + Q P & = O P \\ O Q + 6 & = 17 \\ O Q & = 17 - 6 \\ O Q & = 11 \end{aligned}$
Example 5
::OP=OPO6=17O17-6O11例5

What is the distance between the two points shown below?
::下面列出的两点之间的距离是多少?

Because this line is horizontal, look at the change in the $\begin{aligned} x - \end{aligned}$ coordinates.
::因为这条线是水平的,所以请看 x - 坐标的变化。

$\begin{aligned} | (- 4) - 3 | = | - 7 | = 7 \end{aligned}$

The distance between the two points is 7 units.
::两点之间的距离是7个单位。

Review
::回顾

For 1-4, use the ruler in each picture to determine the length of the line segment.
::1-4,使用每幅图片的标尺确定线段的长度。

Make a sketch of $\begin{aligned} \bar{B T} \end{aligned}$ , with $\begin{aligned} A \end{aligned}$ between $\begin{aligned} B \end{aligned}$ and $\begin{aligned} T \end{aligned}$ .
::做一张BT的草图加上B和T之间的A

If $\begin{aligned} O \end{aligned}$ is in the middle of $\begin{aligned} \bar{L T} \end{aligned}$ , where exactly is it located? If $\begin{aligned} L T = 16 c m \end{aligned}$ , what is $\begin{aligned} L O \end{aligned}$ and $\begin{aligned} O T \end{aligned}$ ?
::如果O在LT中间,它的确切位置在哪里?如果LT=16厘米,什么是LO和OT?

For three collinear points, $\begin{aligned} A \end{aligned}$ between $\begin{aligned} T \end{aligned}$ and $\begin{aligned} Q \end{aligned}$ :

Draw a sketch.
::画个草图

Write the Segment Addition Postulate for your sketch.
::写下您草图的片段附加假设。

If $\begin{aligned} A T = 10 i n \end{aligned}$ and $\begin{aligned} A Q = 5 i n \end{aligned}$ , what is $\begin{aligned} T Q \end{aligned}$ ?
::如果AT=10和AQ=5,什么是TQ?

::对于三个圆线点, A 在 T 和 Q 之间 : 绘制一个草图。为您的草图写线段附加假设。如果 AT= 10 和 AQ 5, 什么是 TQ ?

For three collinear points, $\begin{aligned} M \end{aligned}$ between $\begin{aligned} H \end{aligned}$ and $\begin{aligned} A \end{aligned}$ :

Draw a sketch.
::画个草图

Write the Segment Addition Postulate for your sketch.
::写下您草图的片段附加假设。

If $\begin{aligned} H M = 18 c m \end{aligned}$ and $\begin{aligned} H A = 29 c m \end{aligned}$ , what is $\begin{aligned} A M \end{aligned}$ ?
::如果HM=18厘米和HA=29厘米,什么是AM?

::对于三个圆线点, M 在 H 和 A 之间: 绘制一个草图。为您的草图写线段附加假设。如果 HM= 18 cm 和 HA= 29 cm, 什么是AM ?

For three collinear points, $\begin{aligned} I \end{aligned}$ between $\begin{aligned} M \end{aligned}$ and $\begin{aligned} T \end{aligned}$ :

Draw a sketch.
::画个草图

Write the Segment Addition Postulate for your sketch.
::写下您草图的片段附加假设。

If $\begin{aligned} I T = 6 c m \end{aligned}$ and $\begin{aligned} M T = 25 c m \end{aligned}$ , what is $\begin{aligned} A M \end{aligned}$ ?
::如果IT=6厘米和MT=25厘米,什么是AM?

::对于三个圆线点, 我介于 M 和 T 之间: 绘制一个草图。为您的草图写下线段补充假设。如果 IT= 6 cm 和 MT= 25 cm, 什么是AM ?

Make a sketch that matches the description: $\begin{aligned} B \end{aligned}$ is between $\begin{aligned} A \end{aligned}$ and $\begin{aligned} D \end{aligned}$ . $\begin{aligned} C \end{aligned}$ is between $\begin{aligned} B \end{aligned}$ and $\begin{aligned} D \end{aligned}$ . $\begin{aligned} A B = 7 c m, A C = 15 c m \end{aligned}$ , and $\begin{aligned} A D = 32 c m \end{aligned}$ . Find $\begin{aligned} B C, B D \end{aligned}$ , and $\begin{aligned} C D \end{aligned}$ .
::绘制符合描述的草图:B介于A和D之间。B介于A和D之间。AB=7厘米,AC=15厘米,AD=32厘米。找到BC、BD和CD。

Make a sketch that matches the description: $\begin{aligned} E \end{aligned}$ is between $\begin{aligned} F \end{aligned}$ and $\begin{aligned} G \end{aligned}$ . $\begin{aligned} H \end{aligned}$ is between $\begin{aligned} F \end{aligned}$ and $\begin{aligned} E \end{aligned}$ . $\begin{aligned} F H = 4 i n, E G = 9 i n \end{aligned}$ , and $\begin{aligned} F H = H E \end{aligned}$ . Find $\begin{aligned} F E, H G \end{aligned}$ , and $\begin{aligned} F G \end{aligned}$ .
::绘制符合描述的草图: E介于F和G之间。 H介于F和E. FH=4之间, EG=9英寸, FH=HE。查找FE、 HG和FG。

For 12 and 13, Suppose $\begin{aligned} J \end{aligned}$ is between $\begin{aligned} H \end{aligned}$ and $\begin{aligned} K \end{aligned}$ . Use the Segment Addition Postulate to solve for $\begin{aligned} x \end{aligned}$ . Then find the length of each segment.
::对于 12 和 13 , 假设 J 在 H 和 K 之间。使用 part + 假设来解决 x 。然后找到每个段的长度。

$\begin{aligned} H J = 4 x + 9, J K = 3 x + 3, K H = 33 \end{aligned}$
::HJ=4x+9, JK=3x+3, KH=33

$\begin{aligned} H J = 5 x - 3, J K = 8 x - 9, K H = 131 \end{aligned}$
::HJ=5x-3,JK=8x-9,KH=131

For 14-17, determine the vertical or horizontal distance between the two points.
::对于14-17,确定两个点之间的垂直或水平距离。

Make a sketch of: $\begin{aligned} S \end{aligned}$ is between $\begin{aligned} T \end{aligned}$ and $\begin{aligned} V \end{aligned}$ . $\begin{aligned} R \end{aligned}$ is between $\begin{aligned} S \end{aligned}$ and $\begin{aligned} T \end{aligned}$ . $\begin{aligned} T R = 6, R V = 23 \end{aligned}$ , and $\begin{aligned} T R = S V \end{aligned}$ .
::草图如下:S介于T和V之间;R介于S和T.TR=6、RV=23和TR=SV之间。

Find $\begin{aligned} S V, T S, R S \end{aligned}$ and $\begin{aligned} T V \end{aligned}$ from #18.
::找到18号街的SV、TS、RS和电视

For $\begin{aligned} \bar{H K} \end{aligned}$ , suppose that $\begin{aligned} J \end{aligned}$ is between $\begin{aligned} H \end{aligned}$ and $\begin{aligned} K \end{aligned}$ . If $\begin{aligned} H J = 2 x + 4, J K = 3 x + 3 \end{aligned}$ , and $\begin{aligned} K H = 22 \end{aligned}$ , find $\begin{aligned} x \end{aligned}$ .
::就香港而言,假设J介于H和K之间。如果HJ=2x+4、JK=3x+3和KH=22,请找到x。

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

两点之间的距离

Measuring Distance Between Two Points ::测量两点之间的距离

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

O Q + Q P = O P O Q + 6 = 17 O Q = 17 − 6 O Q = 11 Example 5 ::OP=OPO6=17O17-6O11例5

Review ::回顾

Review (Answers) ::回顾(答复)

Measuring Distance Between Two Points
::测量两点之间的距离

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

$\begin{aligned} O Q + Q P & = O P \\ O Q + 6 & = 17 \\ O Q & = 17 - 6 \\ O Q & = 11 \end{aligned}$
Example 5
::OP=OPO6=17O17-6O11例5

Review
::回顾

Review (Answers)
::回顾(答复)