8.15 类似数字的未知计量

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  类似数字的未知计量

  

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  Shelby's father is a former Olympic swimmer. He loves swimming, but he cannot build an Olympic sized swimming pool in his backyard because he does not have the space . He decides to build a smaller pool that is similar to the dimensions of an Olympic pool. An Olympic pool has a length of 50 meters and a width of 25 meters. His pool has a length of 25 meters. What is the width of his pool?
  ::谢尔比的父亲是前奥林匹克游泳运动员。他喜欢游泳,但他不能在他的后院建造一个奥林匹克规模的游泳池,因为他没有空间。他决定建造一个与奥林匹克游泳池尺寸相似的小型游泳池。奥林匹克游泳池的长度为50米,宽度为25米。他的游泳池长25米。他的游泳池宽度是多少?

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  In this concept, you will learn how to find the unknown measures of .
  ::在这一概念中,你将学会如何找到未知的衡量标准。

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  Finding Unknown Measures of Similar Figures
  ::寻找类似数字的未知度量

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  If you know the length of a side in one figure, you can use the scale factor to find the measure of the corresponding side in a similar figure. Let’s see how this works.
  ::如果您知道一个数字中侧面的长度, 您可以使用比例系数在类似数字中找到相应侧面的度量。 让我们来看看这个方法如何运作 。

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  Side  $\begin{array}{r}a\end{array}$ in triangle $\begin{array}{r}ABC\end{array}$ corresponds to side $\begin{array}{r}x\end{array}$ in the smaller triangle $\begin{array}{r}XYZ\end{array}$ . Side $\begin{array}{r}x\end{array}$ is 4 meters long and the scale factor is 6. What is the measure of side $\begin{array}{r}a\end{array}$ ?
  ::三角形 ABC 的侧面a 与小三角形 XYZ 的侧面 x 对应。 侧面x 长4米, 比例系数为 6 。 侧面a 的度量是多少?

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  You have been told that two sides, $\begin{array}{r}a\end{array}$ and $\begin{array}{r}x\end{array}$ , correspond in a small triangle and a large one. Side $\begin{array}{r}x\end{array}$ is 4 meters long, and the scale factor tells you that side $\begin{array}{r}a\end{array}$ will be six times longer. Let’s write this out and solve.
  ::你被告知,两边(a和x)在一个小三角形和一个大三角形中对应。 方x长4米,比例系数告诉你,侧a将长6倍。让我们写出来解决。

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  $$\begin{array}{rl}\text{side}x\times \text{scale factor}& =\text{side}a\\ 4\times 6& =\text{side}a\\ 24m& =\text{side}a\end{array}$$

  
  ::平面 xxx 比例因数= 平面 a4x6= 平面 a24 m= 平面 a

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  Side  $\begin{array}{r}a\end{array}$ must have a length of 24 meters.
  ::a 侧面必须长24米

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  You can check by setting up a ratio that compares the lengths of the two sides. If the scale factor is 6, then your work is accurate.
  ::您可以设置一个比对比例来比较两边的长度。如果比例系数为6,那么您的工作是准确的。

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  $\begin{array}{r}\frac{a}{x}=\frac{24}{4}=6\end{array}$
  ::ax=244=6

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  Sometimes, you can figure out missing side lengths by looking at the given measures. Always look at the diagram of the figures and see if you can determine the missing length without measuring.
  ::有时,您可以通过查看给定的测量方法来找出缺失的侧边长度。 总是要查看数字的图表, 看看您能否在不进行测量的情况下确定缺失的长度 。

  

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  Look at these two . First, look and see if you can figure out the relationship between the two figures. To do this, you compare the side lengths of each part of the two figures.
  ::看看这两个数字。 首先, 看看能不能找出这两个数字之间的关系。 要做到这一点, 您可以比较两个数字的侧边长度 。

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  You need to figure out the measurement of side $\begin{array}{r}GH\end{array}$ in the second rectangle .
  ::你需要弄清楚第二个矩形中 侧GH的测量方法

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  You can see that the measurements in the second rectangle are half as big as the measurements in the first. Also, you know that the opposite sides of a rectangle are congruent . Therefore, the missing side length is 4.
  ::您可以看到第二个矩形的测量值与第一个矩形的测量值大一半。 另外, 您也知道矩形的对面是相同的。 因此, 缺失的侧长是 4 。

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  Similar figures that are related by a scale factor are often seen in maps, architectural blueprints and diagrams . In most of these cases, the scale factor is given so that you know how to enlarge the items in the drawing to their real sizes. Take a look at the floor plan below. It shows where the furniture is located in a living room.
  ::在地图、建筑蓝图和图表中,经常看到与比例因素有关的类似数字,在多数情况下,给出比例因素是为了让你知道如何将绘图中的项目扩大至其实际大小。请看看下面的平面图。它显示家具位于一个客厅。

  

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  It tells you that one inch in the drawing is equal to two feet in actual size. Therefore, if you know the size in inches of any object in the floor plan, you can find its actual size in feet. Let’s give it a try.
  ::它告诉你,绘图中有一英寸等于实际大小的两英尺。 因此,如果你知道地板图中任何物体的大小,你可以在脚中找到实际大小。 让我们试一试。

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  Find the sofa on the floor plan then find its length in inches. The sofa in the floor plan is 2 inches long. This is like knowing the length of one side in a similar figure. Now use the scale factor as you would to find the length of the corresponding side in a similar figure (in this case the “corresponding side” is the actual sofa). Simply multiply the length you know by the scale factor:
  ::查找地板图上的沙发,然后找到它的长度英寸。 地板图中的沙发长2英寸。 这就像知道类似数字中一边的长度一样。 现在使用比例系数来查找类似数字中对应一边的长度( 在这种情况下, “ 对应边” 是实际的沙发)。 简单地乘以您所知道的长度 :

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  $$\begin{array}{rl}\text{sofa drawing}\times \text{scale factor}& =\text{actual sofa size}\\ 2inches\times 2& =4feet\end{array}$$

  
  ::=实际沙发大小2英寸2x2=4英尺

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  The sofa is 4 feet long.
  ::沙发长四英尺

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  Next, let's calculate the real length of the fireplace.Use a ruler to measure the fireplace in the drawing. It is 2.5 inches long. Multiply this by the scale factor to find the length in feet.
  ::接下来,让我们计算壁炉的实际长度。 使用标尺测量绘图中的壁炉。 它长2.5 英寸。 用这个乘以比例系数来找到脚的长度 。

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  $$\begin{array}{rl}\text{fireplace drawing}\times \text{scale factor}& =\text{actual fireplace length}\\ 2.5inches\times 2& =5feet\end{array}$$

  
  ::=实际壁壁长度2.5英寸xx2=5英尺

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  The real length of the fireplace is 5 feet.
  ::壁炉的实际长度是5英尺

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  You can also reverse the process to take an actual size and reduce it.
  ::您也可以将进程倒转,以采取实际的大小并缩小其规模。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Shelby's father and his swimming pool.
  ::早些时候,你得到一个问题 关于谢尔比的父亲 和他的游泳池。

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  He builds a pool that is similar to an Olympic pool. An Olympic pool has a length of 50 meters and a width of 25 meters. If his pool has a length of 25 meters, what is the width of his pool?
  ::他建造了一个类似于奥林匹克游泳池的游泳池,奥林匹克游泳池长50米,宽25米。如果他的游泳池长25米,那么他的游泳池宽度是多少?

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  First, set up an equation that can be used to solve for the width of his pool.
  ::首先,设置一个方程式 用来解决他游泳池宽度的问题

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    $\begin{array}{r}\frac{50}{25}=\frac{25}{x}\end{array}$
  ::5025=25x

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  Next, cross multiply.
  ::下一个,交叉乘数。

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    $\begin{array}{r}50x=625\end{array}$
  ::50x=625

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  Then, divide both sides of the equation by 50.
  ::然后,将等式两侧除以50。

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    $\begin{array}{r}x=\end{array}$ 12.5
  ::x=12.5

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  The answer is that his pool has a width of 12.5 meters.
  ::答案是他的游泳池宽度12.5米

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  Example 2
  ::例2

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  Use the scale factor of the similar figures below to find the measure of $\begin{array}{r}KJ\end{array}$ .
  ::使用以下类似数字的比额表因数来计算 KJ 的尺度。

  

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  First, set up a proportion to solve for the missing side.
  ::首先,设定一个比例 解决失踪的一方。

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  $\begin{array}{r}\frac{KJ}{5}=\frac{6}{4}\end{array}$
  ::KJ5=64

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  This proportion is written so that the corresponding sides form the two ratios of the proportion. You can say that $\begin{array}{r}KJ\end{array}$ is the unknown in this proportion.
  ::此比例是写成的, 以便相应的两边构成比例的两种比例。 您可以说 KJ 是这个比例中未知的 。

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  Next, cross multiply. 
  ::下一个,交叉乘数。

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  $$\begin{array}{rl}KJ\times 4& =4KJ\\ 5\times 6& =30\\ 4KJ& =30\end{array}$$

  
  ::KJx4=4KJ5x6=304KJ=30

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  Then solve the equation for $\begin{array}{r}KJ\end{array}$ by dividing both sides of the equation by 4.
  ::然后通过将方程两侧除以 4 来解开KJ的方程。

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  $$\begin{array}{rl}30÷4& =7.5\\ KJ& =7.5\end{array}$$

  
  ::304=7.5KJ=7.5

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  The answer is that the side length of $\begin{array}{r}KJ\end{array}$ is 7.5.
  ::答案是KJ的侧长是7.5。

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  Example 3
  ::例3

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  Chris is making a drawing of his school and the grounds around it. The basketball court is 75 feet long and 40 feet wide. If Chris uses a scale factor in which 1 inch equals 10 feet, what should the dimensions of the basketball court be in his drawing?
  ::克里斯正在画他的学校及其周围的场地。 篮球场有75英尺长,40英尺宽。 如果克里斯使用一个1英寸等于10英尺的尺度系数,那么篮球场的尺寸应该如何?

  

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  First, write the information that you know.
  ::首先,写下你知道的信息。

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  The actual size of the basketball court and the scale factor Chris is using for his drawing.
  ::篮球场的实际规模 和比例系数克里斯正在使用 用于他的绘画。

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  Next, set up an equation to find the length Chris should draw. 
  ::接下来,设置一个方程 以找到克里斯应该画的长度。

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  $$\begin{array}{rl}\text{drawing length}\times \text{scale factor}& =\text{actual basketball court length}\\ \text{drawing length}\times 10& =75feet\\ \text{drawing length}& =75÷10\\ \text{drawing length}& =7.5inches\end{array}$$

  
  ::=实际篮球场长拖长度x10=75英尺长=7510拖长=7.5英寸

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  The length of the basketball court in Chris’s drawing should be 7.5 inches.
  ::Chris画图时篮球场的长度应该为7.5英寸。

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  Then, use the same process to find the width Chris should draw.
  ::然后,使用同样的程序来寻找Chris应该画的宽度。

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  $$\begin{array}{rl}\text{drawing width}\times \text{scale factor}& =\text{actual basketball court width}\\ \text{drawing width}\times 10& =40feet\\ \text{drawing width}& =40÷10\\ \text{drawing width}& =4inches\end{array}$$

  
  ::宽度x比例系数=实际篮球场宽度宽度x10=40英尺宽度=4010度宽度=4英寸

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  The answer is that Chris should represent the basketball court as a 7.5 by 4 inch rectangle on his drawing.
  ::答案是克里斯应该代表篮球场 在他的绘画上以7.5乘4英寸的矩形 代表篮球场

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  Example 4
  ::例4

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  Solve for $\begin{array}{r}x\end{array}$ .
  ::解决x。

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  $\begin{array}{r}\frac{3}{4}=\frac{x}{12}\end{array}$
  ::34=12 =34=12

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  First, cross multiply.
  ::首先,交叉乘数。

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  4 $\begin{array}{r}x\end{array}$ = 36
  ::4x=36

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  Next, divide both sides of the equation by 4.
  ::其次,将等式的两侧除以4。

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  $\begin{array}{r}x\end{array}$  = 9
  ::x = 9

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  Then, state the solution.
  ::然后,说出解决办法。

  9

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  The answer is that  $\begin{array}{r}x\end{array}$ equals 9.
  ::答案是 x 等于 9 。

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  Example 5
  ::例5

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  Solve for $\begin{array}{r}x\end{array}$ .
  ::解决x。

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  $\begin{array}{r}\frac{3}{6}=\frac{1}{x}\end{array}$
  ::36=1x

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  First, cross multiply.
  ::首先,交叉乘数。

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  3 $\begin{array}{r}x\end{array}$ = 6
  ::3x=6

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  Next, divide both sides of the equation by 3.
  ::接下来,将方程两侧除以3。

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  $\begin{array}{r}x\end{array}$  = 2
  ::x=2x=2

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  Then, state the solution.
  ::然后,说出解决办法。

  2

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  The answer is that  $\begin{array}{r}x\end{array}$ equals 2.
  ::答案是 x 等于 2 。

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  Review
  ::回顾

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  Solve each problem.
  ::解决每一个问题。

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  1. Side $\begin{array}{r}m\end{array}$ in triangle $\begin{array}{r}LMN\end{array}$ corresponds to side $\begin{array}{r}c\end{array}$ in the smaller triangle $\begin{array}{r}BCD\end{array}$ . Side $\begin{array}{r}m\end{array}$ is 12 cm long and the scale factor from the larger to the smaller triangle is $\begin{array}{r}\frac{1}{4}.\end{array}$  What is the measure of side $\begin{array}{r}c\end{array}$ ?
     ::三角形 LMN 的侧边 m 对应小三角形 BCD 的 c 侧 。 侧 m 长 12 厘米, 从大三角形到小三角形的尺度因子是 14 。 侧 c 的量度是 14 ?
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  2. Side $\begin{array}{r}q\end{array}$ in triangle $\begin{array}{r}PQR\end{array}$ corresponds to side $\begin{array}{r}y\end{array}$ in the smaller triangle $\begin{array}{r}XYZ\end{array}$ . Side $\begin{array}{r}y\end{array}$ is 8 inches long and the scale factor from the smaller to the larger triangle is 7. What is the measure of side $\begin{array}{r}q\end{array}$ ?
     ::三角形 PQR 中的侧方q 与小三角形 XYZ 中的侧 y 对应。 侧 y 长8 英寸, 从小三角形到大三角形的尺度因子是 7 。 侧 q 的量度是多少?

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  Solve each proportion for the missing side length.
  ::为缺失的侧长度解决每个比例 。

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  3. $\begin{array}{r}\frac{7}{10}=\frac{x}{30}\end{array}$
     ::710=x30
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  4. $\begin{array}{r}\frac{1.5}{3}=\frac{x}{6}\end{array}$
     ::1.53=x6

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  Given the initial side length and the scale factor, find the length of the corresponding side.
  ::考虑到最初的副长度和比额表系数,找到相应边的长度。

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  5. $\begin{array}{r}\frac{1}{3}\end{array}$ inch, scale factor 4
     ::13英寸,比额表因数4
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  6. $\begin{array}{r}1\frac{3}{5}\end{array}$ feet, scale factor 5
     ::135英尺,比额表因数5
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  7. $\begin{array}{r}\frac{9}{3}\end{array}$ cm, scale factor 3
     ::93厘米,比额表因数3
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  8. Find the scale factor of the similar figures below and then use it to find the measure of $\begin{array}{r}LO\end{array}$ .
     ::查找以下类似数字的尺度系数,然后用它来找到联络处的尺度。

  

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  9. Find the scale factor of the similar figures below to find the measure of $\begin{array}{r}JK\end{array}$ .
     ::查找以下类似数字的比额表因数以找到JK的尺度。

  

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  Use the map below and a ruler to answer the questions that follow.
  ::使用下面的地图和一个标尺回答下面的问题。

  

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  10. How far does Delia live from her school?
      ::Delia离学校有多远?
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  11. How far is it from the library to the park?
      ::从图书馆到公园有多远?
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  12. How far does Delia live from City Hall?
      ::Delia离市政厅有多远?
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  13. Delia drew another point to show the police station on her map. She drew it 1.5 inches away from the City Hall. What is the actual distance between the police station and City Hall?
      ::Delia画了另一个点,在地图上向警察局展示她的地图,她从市政厅抽出1.5英寸。
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  14. How far does Delia live from the park?
      ::Delia离公园有多远?
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  15. How far does she live from the library?
      ::她离图书馆有多远?
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  16. What is the farthest that Delia will travel to any one item on her map?
      ::Delia去地图上任何一件物品最远的地方是什么?

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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 。

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  Resources
  ::资源