2.5 界定相似性

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  界定相似性

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  Thales and the Pyramids
  ::赛马人和金字马,

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  Engineering Question: How is it possible to measure the height of something too tall to climb easily, or something with a height that  just can't be measured directly, such as the Great Pyramids of Giza in Egypt ? Clearly there is no way to drop a measuring tape from the peak down through the solid rock to the base directly beneath it!  What tools might you use to make such tall measurements? Are there any?
  ::工程问题:如何能测量高到无法轻易攀登的高度,或者高到无法直接测量的高度,比如埃及的吉萨大金字塔?显然无法将测量磁带从山顶从坚固的岩石向下方的底部投下。你可以用什么工具来进行如此高的测量?有没有?

  

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  The Pyramids of Giza
  ::吉萨的金字塔

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  Approximately 400 years before Eratosthenes, the ancient Greek mathematician Thales of Miletus also studied in Egypt. Like Eratosthenes, Thales studied the techniques of Egyptian surveyors and engineers. One of the most well-known tales of Thales’ adventures in Egypt  tells  of  how he was able to determine the height of the Great Pyramid of Giza. Although the Great Pyramid had been around for over 2,000 years, no one knew the height of the pyramid. U sing only the shadows cast by the pyramids and his knowledge of similarity, Thales was able to estimate the height of the pyramid to be approximately 147 meters or 481 feet tall. Later in this lesson ,  you will explore how Thales was able to use his understanding of similarity to find this number.
  ::古希腊数学家麦利图斯(Miletus)的塔雷斯(Thales)也在埃及学习。与埃拉托斯(Eratosthenes)一样,塔勒斯研究埃及测量员和工程师的技巧。在埃及,塔勒斯最著名的塔勒斯冒险故事之一讲述了他如何能够确定吉萨大金字塔的高度。 尽管大金字塔已经存在了2000多年,但没有人知道金字塔的高度。 仅利用金字塔所投下的阴影和他所了解的相似之处,塔勒斯就能够估计金字塔的高度约为147米或481英尺。 稍后,你将探索塔勒斯如何利用他对相似之处的理解找到这个数字。

   

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  Defining Similarity
  ::界定相似性

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  The lesson   discussed  how dilation is used to animate an object so that it appears to  move towards or away from the screen . When an image is dilated , what aspects of the image change? Do the angles of the shapes change? Do the side lengths of the shapes change?
  ::讨论过的课程如何使用放大法来动画一个对象,使其向屏幕或向屏幕移动。 当图像膨胀时, 图像的哪些方面会变化? 形状的角度会改变吗? 形状的侧边长度会改变吗 ?

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  Recall that when an image is dilated, this causes the side lengths to change, but the angles remain the same. Two shapes that can be produced by dilating one to obtain the other are called similar shapes.
  ::回顾当图像膨胀时,这会导致侧长变化,但角度保持不变。 两种形状被称为相似形状。 两种形状可以通过放大一个来获取另一个形状而产生。

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  Similarity means having equal and proportional corresponding sides . For example, if shape $\begin{align*}A\end{align*}$  is doubled in size, then the resulting shape $\begin{align*}A’ \end{align*}$  will have corresponding angles that are the same as those in  $\begin{align*} A\end{align*}$  and every side length  of   $\begin{align*}A’ \end{align*}$ will be double the corresponding side length of $\begin{align*}A\end{align*}$ .
  ::相似是指具有相同和成比例的对应边。例如,如果形状A的大小翻了一番,则所产生的形状AAA将具有与A相同的相应角度,而AA的每一侧长度将是A的对应侧长度的两倍。

  

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  Two similar triangles. All corresponding angles are congruent, and all corresponding sides are proportional (2:1, in this case)
  ::两个相似的三角形。所有相应的角度都是相似的,所有对应的两边都是比例的(在此情况下为 2:1 )

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  The idea of similarity is different from congruence because congruent images must be the same size and shape while similar shapes only need to have the same shape . The notation for similarity is similar to the notation used to express congruence.
  ::相似性的概念与一致性不同,因为一致图像的大小和形状必须相同,而相似的形状只需要相同的形状。 相似性的符号类似于用来表示一致性的符号。

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  Instead of the symbol $\begin{align*}\cong, \end{align*}$  which is used to show congruence, use the symbol $\begin{align*}\sim\end{align*}$  to show similarity. To show that triangle  $\begin{align*}ABC\end{align*}$  and triangle $\begin{align*}XYZ\end{align*}$  are similar, use the notation  $\begin{align*}ΔABC \sim ΔXYZ.\end{align*}$  
  ::用于显示一致性的符号 {} , 使用符号 { } 来显示相似性。 要显示三角 ABC 和三角 XYZ 相似, 请使用标记 \ begin{ align} ABC\ sim \ XYZ.\ end{ ALign} 。

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  Use the interactive below to determine whether the two shapes are congruent, similar or neither.
  ::使用下面的交互功能来确定两种形状是否一致、相似或两者都不相同。

   

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  Measuring the Great Pyramid of Giza
  ::测量吉萨大金字塔

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  Thales, who understood the concept of similarity, realized that because the sun hit objects in the same general area at the same angle ,   every  object and  its shadow create similar triangles . Thales compared the shadow of a stick with the shadow of the pyramid. Thales used the unit of a cubit to obtain these measurements. A cubit is a commonly used ancient unit and is equal to the distance from your elbow to your fingertips. In modern day, a cubit is considered to be 18 inches. Thales used a stick which was 6 cubits tall and  stood it on the ground so that it was pointing directly upward. The shadow of the stick was 4 cubits long, and the shadow of the pyramid was 214 cubits long.
  ::了解相似概念的泰勒斯认识到,由于太阳从同一角度击打同一大区物体,每个物体及其阴影都形成类似的三角形。泰勒斯将棍子的阴影与金字塔的阴影进行比较。泰勒斯使用一个肘子单位来进行测量。一个肘子是常用的古代单元,相当于从肘部到手指部的距离。在现代,一只肘子被认为是18英寸。泰勒斯使用一根6肘高的棍子,并站在地上,直接指向上。棍子的阴影是4肘长,金字塔的阴影是214肘长。

  

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  The shadow from Thales's stick and the shadow from the pyramid of Giza.
  ::泰勒斯棍子的影子 吉萨金字塔的影子

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  Thales knew that the triangles were similar since the angle between the sun rays (the hypotenuse) and the ground was the same, and both triangles had 90 degree angles. Since every triangle has a total of 180 degrees, the upper angle must be the same also!
  ::泰勒斯知道三角形是相似的,因为太阳射线(天线)和地面之间的角是相同的,而两个三角形都有90度角。由于每个三角形的总度为180度,上角也必须相同!

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  Since the triangles were similar, that meant that their corresponding sides we re proportional.
  ::由于三角形相似,这意味着它们对应的两边是成比例的。

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  Thales  used the following proportion and cross-multiplication to determine the height of the pyramid:
  ::泰勒斯使用以下比例和交叉乘法来确定金字塔的高度:

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  $$\begin{align*}\frac{\text{height of the stick}}{\text{length of the stick's shadow}} &= \frac{\text{height of the pyramid}}{\text{length of the pyramid's shadow}}\\ \\ \frac{6}{4}&=\frac{x}{214}\\ \\ 4\cdot x &= 6\cdot 214\\ 4x &= 1284\\ x &= 321\text{ cubits}\end{align*}$$
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  321 cubits is approximately 147 meters or 481 feet tall. The accepted measure of the height of the Great Pyramid today is about 455 feet, but erosion has significantly reduced the height from its original measure.
  ::321公尺约147公尺或481公尺,大金字塔的高度目前为455公尺,但侵蚀已大大降低其原有的高度。

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  Use the interactive below to practice finding missing sides and angles in similar shapes.
  ::使用以下互动方式,以类似形状查找缺失的侧面和角度。

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   Summary
  ::摘要

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  • Similar shapes have congruent corresponding angles and proportional corresponding sides.
    ::相似的形状具有相似的对应角度和比例对应的边。
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  • The symbol to show similarity is  $\begin{align*}\sim.\end{align*}$
    ::显示相似性的符号是\ begin{ align_ simm.\ end{ aliign} 。
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  • To find side lengths of similar shapes,  set up a proportion between corresponding sides and cross-multiply.
    ::为了找到相似形状的侧长,在对应面和交叉倍数之间设定比例。