7.5 间接计量

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  间接计量

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  Indirect Measurement
  ::间接计量

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  An application of similar triangles is to measure lengths indirectly. You can use this method to measure the width of a river or canyon or the height of a tall object. The idea is that you model a situation with similar triangles and then use proportions to find the missing measurement indirectly.
  ::类似三角形的应用是间接测量长度。 您可以使用这个方法测量河流或峡谷的宽度或高物体的高度。 想法是您用相似三角形建模一个情况, 然后用比例来间接找到缺失的测量值 。

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  What if you were standing next to a building and wanted to know how tall the building was? How could you use your own height and the length of the shadows cast by you and the building to determine the building's height?
  ::如果你站在大楼旁边,想知道大楼有多高呢?你怎么能用自己的身高和阴影的长度来决定大楼的高度?

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  Examples
  ::实例

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  For Examples 1, 2, and 3, use the following information:
  ::关于例1、2和3,请使用以下信息:

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  In order to estimate the width of a river, the following technique can be used. Use the diagram .
  ::为了估计河流的宽度,可以使用以下技术。使用图表。

  

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  Place three markers, $\begin{align*}O, C,\end{align*}$ and $\begin{align*}E\end{align*}$ on the upper bank of the river. $\begin{align*}E\end{align*}$ is on the edge of the river and $\begin{align*}\overline {OC} \perp \overline{CE}\end{align*}$ . Go across the river and place a marker, $\begin{align*}N\end{align*}$ so that it is collinear with $\begin{align*}C\end{align*}$ and $\begin{align*}E\end{align*}$ . Then, walk along the lower bank of the river and place marker $\begin{align*}A\end{align*}$ , so that $\begin{align*}\overline{CN} \perp \overline{NA}\end{align*}$ . $\begin{align*}OC = 50\end{align*}$ feet, $\begin{align*}CE = 30\end{align*}$ feet, $\begin{align*}NA = 80\end{align*}$ feet.
  ::在河上上岸设置三个标记,O、C和E。E在河边,C.C.C.,C.C.C.,C.C.C.,C.C.C.,C.C.C.C.,C.C.C.C.C.C.,C.C.C.C.C.C.,C.C.C.C.C.,C.C.C.C.,C.C.C.C.,C.C.C.C.,C.E.C.C.C.,C.E.C.C.C.C.C.,C.E.C.C.C.C.,C.A.A.,C.C.C.C.C.A.A.,C.C.C.C.A.A.A.A.A.A.A.A.,C.C.C=50英尺,C.C.C.C.C.C.C.C.C.C.C.C.和C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.和C.C.A.A.A.A.A.A.A.

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  Example 1
  ::例1

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  Is $\begin{align*}\triangle OCE \sim \triangle ANE?\end{align*}$ How do you know?
  ::欧克安,你怎么知道的?

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  Yes. $\begin{align*}\angle{C} \cong \angle{N}\end{align*}$ because they are both right angles . $\begin{align*}\angle{OEC} \cong \angle{AEN}\end{align*}$ because they are . This means $\begin{align*}\triangle OCE \sim \triangle ANE\end{align*}$ by the AA Similarity Postulate .
  ::是的,因为两者都是正确的角度。这意味着AA相似的假设。

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  Example 2
  ::例2

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  Is $\begin{align*}\overline {OC} \| \overline {NA}?\end{align*}$ How do you know?
  ::你怎么会知道的?

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  Since the two triangles are similar , we must have $\begin{align*}\angle{EOC} \cong \angle{EAN}\end{align*}$ . These are . When alternate interior angles are congruent then lines are parallel , so $\begin{align*}\overline {OC} \| \overline {NA}\end{align*}$ .
  ::因为这两个三角形相似, 我们必须有“ EOC ” 。 这些是 。 当其它内部角度相似时, 线线是平行的, 所以是 OC NA 。

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  Example 3
  ::例3

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  What is the width of the river? Find $\begin{align*}EN\end{align*}$ .
  ::河的宽度是多少?

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  Set up a proportion and solve by cross-multiplying.
  ::设定一个比例, 并通过交叉倍增解决 。

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  $$\begin{align*}\frac{30 ft}{EN}&=\frac{50 ft}{80 ft}\\ 50(EN)&=2400 \\ EN&=48\end{align*}$$
  ::30ftEN=50ft80ft50(EN)=2400EN=48

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  The river is $\begin{align*}48\end{align*}$ feet wide.
  ::河宽48英尺

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  Example 4
  ::例4

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  A tree outside Ellie’s building casts a 125 foot shadow. At the same time of day, Ellie casts a 5.5 foot shadow. If Ellie is 4 feet 10 inches tall, how tall is the tree?
  ::艾莉大楼外的一棵树投下了125英尺的阴影。 当天,艾莉投下了5.5英尺的阴影。 如果艾莉高4英尺10英寸,那树高多少?

  

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  To solve, start by drawing a picture. We see that the tree and Ellie are parallel, so the two triangles are similar.
  ::要解答,首先绘制一张图片。我们看到树和Ellie是平行的,所以两个三角形是相似的。

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  $$\begin{align*}\frac{4 \ ft, 10 \ in}{x } = \frac{5.5 \ ft}{125 \ ft}\end{align*}$$
  ::4英尺10英寸=5.5英尺125英尺

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  The measurements need to be in the same units. Change everything into inches and then we can cross multiply.
  ::测量需要在同一单位内。 将一切改变成英寸, 然后我们就可以交叉乘数。

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  $$\begin{align*}\frac{58 \ in}{x } &= \frac{66 \ in}{1500 \ in}\\ 87000 &= 66x\\ x & \approx 1318. \overline{18} \ in \ \text{or} \ 109.85 \ ft\end{align*}$$
  ::87000=66xx=1318 18英寸或10985英尺

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  Example 5
  ::例5

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  Cameron is 5 ft tall and casts a 12 ft shadow. At the same time of day, a nearby building casts a 78 ft shadow. How tall is the building?
  ::卡梅伦身高5英尺,身高12英尺,身高12英尺。同时,附近一栋大楼也身高78英尺。这栋大楼有多高?

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  To solve, set up a proportion that compares height to shadow length for Cameron and the building. Then solve the equation to find the height of the building. Let $\begin{align*}x\end{align*}$ represent the height of the building.
  ::要解析, 请设定一个比例, 将 Cameron 和 大楼的高度与 阴影长度比较。 然后解析方程式以找到大楼的高度 。 让 x 代表大楼的高度 。

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  $$\begin{align*} \frac{5 ft}{12 ft}&=\frac{x}{78 ft} \\ 12x&=390 \\ x&=32.5 ft\end{align*}$$
  ::5ft12ft=x78ft12x=390x=32.5ft

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  The building is $\begin{align*}32.5\end{align*}$ feet tall.
  ::这栋大楼有32.5英尺高

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  Review
  ::回顾

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  The technique from the guided practice section was used to measure the distance across the Grand Canyon. Use the picture below and $\begin{align*}OC = 72 \ ft , CE = 65 \ ft\end{align*}$ , and $\begin{align*}NA = 14,400 \ ft\end{align*}$ for problems 1 - 3.
  ::指导实践部分的技术用于测量大峡谷之间的距离,用下文和OC=72平方英尺、CE=65平方英尺和NA=14 400平方英尺来测量问题1-3。

  

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  1. Find $\begin{align*}EN\end{align*}$ (the distance across the Grand Canyon).
  ::1. 寻找EN(大峡谷之间的距离)。

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  2. Find $\begin{align*}OE\end{align*}$ .
  ::2. 寻找OE。

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  3. Find $\begin{align*}EA\end{align*}$ .
  ::3. 寻找EA。

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  4. Mark is 6 ft tall and casts a 15 ft shadow. A the same time of day, a nearby building casts a 30 ft shadow. How tall is the building?
  ::4. 马克高6英尺,并投下15英尺的阴影。同一时间,附近一栋大楼投下30英尺的阴影。大楼有多高?

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  5. Karen and Jeff are standing next to each other. Karen casts a 10 ft shadow and Jeff casts an 8 ft shadow. Who is taller? How do you know?
  ::5. Karen和Jeff站在一起,Karen投了10英尺的阴影,Jeff投了8英尺的阴影,谁高?

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  6. Billy is 5 ft 9 inches tall and Bobby is 6 ft tall. Bobby's shadow is 13 feet long. How long is Billy's shadow?
  ::6. Billy身高5英尺9英寸,Bobby身高6英尺,Bobby的影子长13英尺,Billy的影子长多久?

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  7. Sally and her little brother are walking to school. Sally is 4 ft tall and has a shadow that is 3 ft long. Her little brother's shadow is 2 ft long. How tall is her little brother?
  ::7. Sally和她弟弟正在走去上学,Sally身高4英尺高,影子长3英尺长,弟弟身影长2英尺长,小弟弟身高多高?

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  8. Ryan is outside playing basketball. He is 5 ft tall and at this time of day is casting a 12 ft shadow. The basketball hoop is 10 ft tall. How long is the basketball hoop's shadow?
  ::8. 瑞安在外面打篮球,身高5英尺,目前身高12英尺,篮球圈身高10英尺,篮球圈身长多少?

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  9. Jack is standing next to a very tall tree and wonders just how tall it is. He knows that he is 6 ft tall and at this moment his shadow is 8 ft long. He measures the shadow of the tree and finds it is 90 ft. How tall is the tree?
  ::9. Jack站在一棵非常高的树旁,好奇它有多高,他知道他有6英尺高,眼下他的影子长8英尺,他测量了树的阴影,发现树有90英尺高。

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  10. Thomas, who is 4 ft 9 inches tall is casting a 6 ft shadow. A nearby building is casting a 42 ft shadow. How tall is the building?
  ::10. Thomas, 身高4英尺9英寸高,正在投下6英尺高的阴影,附近一栋大楼正在投下42英尺高的阴影,大楼有多高?

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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