间接测量

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Franz has been given the honor of hoisting the flag at his school. He stands next to the flagpole while he hooks the flag. His classmates notice two shadows. Franz is 4 feet tall and his shadow ends at the bench that is 6 feet away from him. The flagpole is 25 feet tall, but the students don't have a direct way to  measure  the  distance  of its shadow. Franz's teacher suggests that the students use indirect measurement to calculate the length of the flagpole's shadow.
::弗朗茨被授予在他的学校升旗的荣誉。 他站在旗杆旁边,同时勾着旗子。 他的同学们注意到了两个阴影。 弗朗茨四英尺高,他的影子在离他六英尺的长凳上结束。 旗杆25英尺高, 但学生们没有直接测量其影子距离的方法。 弗朗茨的老师建议学生们用间接测量法来计算旗杆阴影的长度。

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$\begin{array}{r}\frac{HeightofFranz}{Heightofflagpole}=\frac{ShadowlengthofFranz}{Shadowlengthofflagpole}\end{array}$
::Hi g h t f f f f f f f f f f f n z h y z h f e e = S h a d o f f f f f f y e = S h a d o o e g e y y y z S h h t t t o f f f e y y e y y y y e y y y y e y y e y y e y y y e y y e y y y y y y g h t t t o o f f f f f f f e e y y y y y y S h a a d o d o d o d f e g f f f f f p p o e e

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  $\begin{array}{r}\frac{4^{\prime }}{{25}^{\prime }}=\frac{6^{\prime }}{x}\end{array}$
::4 = 25 = 6 =x

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How can the students determine the length of the flagpole's shadow?
::学生如何决定旗杆阴影的长度?

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In this concept, you will learn how to make an indirect measurement.

::在这个概念中,你会学会如何进行间接计量。

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Indirect Measurement
::间接计量

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You can use the properties of  similar  figures to measure things that are challenging to measure directly. This type of measurement is called  indirect measurement.
::您可以使用类似数字的属性来测量难以直接测量的东西。这类测量称为间接测量。

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Let's look at an example.
::让我们举个例子。

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Jamie’s Dad is six feet tall. Standing outside, his shadow is eight feet long. A tree is next to him. The tree has a shadow that is sixteen feet long. Given these  dimensions , how tall is the tree?
::杰米的爸爸有6英尺高。 站在外边,他的影子有8英尺长。 一棵树在他旁边。 树上有16英尺长的阴影。 鉴于这些尺寸,树有多高?

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Here is a  diagram  to help you.
::这是一张帮助你的图表。

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You can see from the picture that the tree itself is one side of the  triangle , that the shadow length is another side of the triangle and that the  diagonal  from the top of the tree to the top of the shadow forms the  hypotenuse  (the longest side) of the triangle.
::从图中可以看出,树本身是三角形的一面,阴影线是三角形的另一面,从树顶到阴影顶的对角形成三角形的下限(最长的一面)。

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Let's compare it to the shadow of a person.
::让我们把它比作一个人的影子。

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There is a triangle here, too. 
::这里还有一个三角形

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Now let's combine this information.
::现在让我们把这些信息结合起来。

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Jamie’s dad is six feet tall. Standing outside, his shadow is eight feet long. A tree is next to him. The tree has a shadow that is sixteen feet long. Given these dimensions, how tall is the tree?
::杰米的爸爸高6英尺。 站在外面,他的影子长8英尺。 一棵树在他旁边。 树的阴影长16英尺。 鉴于这些尺寸,树有多高?

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To solve this, you have to create two ratios. One will compare the heights of the man and the tree and the other will compare the lengths of the shadows. Together, they will form a  proportion  because  similar triangles  are proportional and you have already seen how triangles are created with people or things and shadows.
::要解决这个问题, 您必须创建两个比例。 一个将比较人的高度和树的高度, 另一个将比较阴影的长度。 它们将形成一个比例, 因为相似的三角形是比例的, 您已经看到三角形是如何与人或事物和阴影一起创建的 。

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Now, fill in the given information.
::现在填入给定的信息 。

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You are looking for the  height  of the tree, so that is where the variable goes.
::您正在寻找树的高度, 所以这就是变量去的地方 。

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Now identify the relationship between the numerical values in the numerator and the denominator.
::现在确定分子和分母中数值之间的关系。

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The denominator is twice the numerator.
::分母是分子的两倍。

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Write and solve an equation to calculate the value of  $\begin{array}{r}x\end{array}$ .
::写入并解析一个公式以计算 x 的值。

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  $\begin{array}{r}x=6\times 2=12\end{array}$
::x = 6 × 2 = 12

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The answer is 12 feet. The tree is 12 feet tall.
::答案是12英尺 树高12英尺

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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 Franz and the flagpole.
::之前,你被问及弗朗茨和旗杆的问题

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Franz is 4 feet tall and his shadow ends at the bench that is 6 feet away from him. Franz's teacher suggests that the students use indirect measurement to calculate the length of the flagpole's shadow.
::弗朗茨四英尺高,他的影子在离他六英尺远的长椅上结束。 弗朗茨的老师建议学生用间接测量法来计算旗杆的影子长度。

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$\begin{array}{r}\frac{HeightofFranz}{Heightofflagpole}=\frac{ShadowlengthofFranz}{Shadowlengthofflagpole}\end{array}$
::Hi g h t f f f f f f f f f f f n z h y z h f e e = S h a d o f f f f f f y e = S h a d o o e g e y y y z S h h t t t o f f f e y y e y y y y e y y y y e y y e y y e y y y e y y e y y y y y y g h t t t o o f f f f f f f e e y y y y y y S h a a d o d o d o d f e g f f f f f p p o e e

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  $\begin{array}{r}\frac{4^{\prime }}{{25}^{\prime }}=\frac{6^{\prime }}{x}\end{array}$
::4 = 25 = 6 =x

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What is the length of the flagpole's shadow?
::旗杆阴影的长度是多少?

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First, find the  cross products .
::首先,找到交叉产品。

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  $\begin{array}{r}4x=6\times 25\end{array}$
::4 x = 6 × 25

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Then, solve for  $\begin{array}{r}x\end{array}$ .
::然后,解决x。

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  $\begin{array}{r}x=\frac{6\times 25}{4}=\frac{150}{4}=37.5\end{array}$
::x = 6 × 25 4 = 150 4 = 37.5

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The answer is that the flagpole's shadow is 37.5 feet long.
::答案是,旗杆的阴影是37.5英尺长。

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Example 2
::例2

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What is the missing side length?
::缺少的侧边长度是什么?

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First, figure out the relationship between the two figures.
::首先,找出这两个数字之间的关系。

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  $\begin{array}{r}\frac{6"}{3"}=\frac{8"}{x}\end{array}$
::" 6 " 3 " =8 " x

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Next, find the cross products.
::接下来,找到交叉产品。

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$\begin{array}{r}6x=8\times 3\end{array}$
::6x=8x3

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Then, solve for  $\begin{array}{r}x\end{array}$ .
::然后,解决x。

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  $\begin{array}{r}x=\frac{8\times 3}{6}=\frac{24}{6}=4\end{array}$
::x = 8 = 8 × 3 6 = 24 6 = 4

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The answer is that  $\begin{array}{r}x=4\end{array}$  inches.
::答案是x=4英寸

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Example 3
::例3

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Are these triangles similar?
::这些三角形相似吗?

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First, check to see if the two triangles are the same size.
::首先,检查两个三角形的大小是否相同。

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No
::否 无

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Next, check to see if the   are  congruent .
::下一位,检查一下是否一致。

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Yes
::是 是

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Then, determine if the two triangles are similar.
::然后,确定这两个三角形是否相似。

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Yes
::是 是

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The answer is that the two triangles are similar.
::答案是这两个三角是相似的。

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Example 4
::例4

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Solve for the missing value.
::解决丢失的值 。

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  $\begin{array}{r}\frac{4^{\prime }}{6^{\prime }}=\frac{{12}^{\prime }}{x}\end{array}$
::4 = 6 = 12 = x

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First, use cross products.
::首先,使用交叉产品。

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  $\begin{array}{r}4x=6\times 12\end{array}$
::4 x = 6 × 12

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Then solve for  $\begin{array}{r}x\end{array}$ .
::然后解答 x。

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  $\begin{array}{r}x=\frac{(6\times 12)}{4}=\frac{72}{4}=18\end{array}$
::x = (6 × 12) 4 = 72 4 = 18

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The answer is that  $\begin{array}{r}x=18\end{array}$ .
::答案是 x = 18 。

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Example 5
::例5

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Use the proportion to indirectly solve for the length of the woman's shadow.
::利用间接解答的比例来弥补妇女影子的长度。

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  $\begin{array}{r}\frac{HeightofWoman}{HeightofTree}=\frac{Shadowlengthofwoman}{Shadowlengthoftree}\end{array}$
::h g h t o f e e e e e e e e e e e e e e e e e e e e e e e e e e e

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  $\begin{array}{r}\frac{5^{\prime }}{{15}^{\prime }}=\frac{x}{{30}^{\prime }}\end{array}$
::5 = 15 = x 30 = = 5 = 15 = = x 30 =

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First, use cross products.
::首先,使用交叉产品。

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$\begin{array}{r}15x=5\times 30\end{array}$
::15 x = 5 × 30

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Then solve for  $\begin{array}{r}x\end{array}$ .
::然后解答 x。

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  $\begin{array}{r}x=\frac{(5\times 30)}{15}=\frac{150}{15}=10\end{array}$
::x = (5 × 30) 15 = 150 15 = 10

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The answer is that  $\begin{array}{r}x=10\end{array}$ . The woman's shadow is 10 feet long.

::答案是 x = 10 。 女人的影子是 10 英尺长 。

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Review
::回顾

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Use indirect measurement and proportions to determine if the figures are similar.
::使用间接计量和比例来确定数字是否相似。

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Use what you have learned about similar triangles and indirect measurement to solve each of the following problems.
::使用您所学到的关于类似三角形和间接测量的知识来解决以下每一个问题。

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5. If a person who is five feet tall casts a shadow that is 8 feet long, how tall is a building that casts a shadow that is 24 feet long?
   ::如果一个5英尺高的人投下8英尺长的阴影,那么一个投下24英尺长的阴影的建筑物有多高?
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6. If a tree stump that is two feet tall casts a shadow that is one foot long, how long is the shadow of a tree that is ten feet at the same time of day?
   ::如果一棵两英尺高的树桩抛下一英尺长的阴影,那么一棵十英尺高的树的阴影还要持续多久呢?
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7. If a 6 foot pole has a shadow that is eight feet long, how tall is a nearby tower that has a shadow that is 16 feet long?
   ::如果一个6英尺高的柱子有8英尺长的阴影,那么一个有16英尺长的阴影的附近的塔子有多高?
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8. If a lifeguard tower is ten feet tall and casts a shadow that is eight feet long, how tall is a person who casts a shadow that is four feet long?
   ::如果一个救生塔有10英尺高, 并投下8英尺长的阴影, 那么一个投下4英尺长的阴影的人有多高?
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9. 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 is 4. What is the measure of side  $\begin{array}{r}c\end{array}$ ?
   ::三角形L M N 的侧边米相当于小三角形B C D 的侧面c。 侧边米长12厘米,比例系数是 4。 侧边的量度是多少?
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10. 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 is 7. What is the measure of side  $\begin{array}{r}q\end{array}$ ?
    ::三角形 P Q R 中的侧方q 对应于小三角形 X Y Z 的侧 y 。 侧 y 长8 英寸, 比例系数是 7 。 侧 Q 的度量是什么?

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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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11. $\begin{array}{r}\frac{3}{4}=\frac{x}{12}\end{array}$
    ::3 4 = x 12
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12. $\begin{array}{r}\frac{3}{6}=\frac{1}{x}\end{array}$
    ::3 6 = 1 x
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13. $\begin{array}{r}\frac{5}{8}=\frac{1}{x}\end{array}$
    ::5 8 = 1 x
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14. $\begin{array}{r}\frac{7}{10}=\frac{x}{30}\end{array}$
    ::7 10 = x 30

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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