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  • Thales Revisited
    ::重审的Thales

    How could you measure the distance of a ship from the shore ?  Would you use any specific tools to accomplish this?
    ::你怎能测量船舶与海岸的距离?你是否使用任何具体工具来完成这个任务?

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    A ship off a shore
    ::离岸的一艘船

    This very problem was faced by Thales of Miletus, the Greek mathematician who famously measured the height of the Great Pyramid using similar triangles .  According to legend, an enemy ship was anchored off the coast of Miletus, Thales’ hometown. The  military needed to know how far the ship was from the shore in order to determine how much time they had to prepare a defense. At the time, the common method for determining the distance from the shore was to estimate based on the size of the ship, but this method did not give a reliable distance. Thales was able to use his knowledge of similar shapes to calculate the distance with much greater accuracy.
    ::这个问题正是Miletus的Thales所面临。 Miletus是希腊数学家,他用类似的三角形测量了大金字塔的高度。 根据传说,一艘敌舰停泊在Thales家乡Miletus海岸附近。 军方需要知道这艘船离海岸有多远才能确定他们需要多少时间准备防御。 当时,确定离海岸距离的常用方法是根据船只的大小来估计距离,但这一方法并不能提供可靠的距离。 泰勒斯能够利用他所了解的类似形状来更精确地计算距离。

     

    Understanding Angle-Angle Similarity
    ::理解角-角相似性

    So how did Thales determine the distance of the ship from the shore?
    ::泰勒斯如何确定船与岸边的距离?

    1. Thales found his starting position ( point B  in the interactive below) by walking along the shore while pointing a stick at the ship until the stick was pointing at a 90° angle to the shore.
      ::Thales发现他的起始位置(B点在下面互动的B点),沿着海岸行走,同时用棍子指着船,直到棍子指向海岸90度角。
    2. Next, he walked along the shoreline and placed   the stick in the ground at point C .
      ::接着,他沿着海岸线走在C点,把棍子放在地上。
    3. Then  he continued walking  along the shoreline  until he  reached  point D .
      ::然后他继续沿着海岸线行走,直到到达D点。
    4. From there, he walked directly away from the sea at a 90° angle until the stick he placed in the ground (point C ) lined up with the ship, point E .
      ::从那里,他以90度角直接离开海洋,直到他把棍子放在地面(C点),与船只E点排成一列。

    This created similar triangles which Thales could use to determine the distance of the ship from the shore. Use the interactive below to explore the similar triangles created by Thales’ path.
    ::这创建了类似三角形, 泰勒斯可以使用这些三角形来确定船舶与海岸的距离。 使用下面的互动来探索由泰勒斯路径创建的类似三角形 。

    INTERACTIVE
    Understanding Thales' Path
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    • Drag the red points to pick points  C  and  D .
      ::拖曳红色点以选择 C 和 D 点 。
    • Press the buttons to see how Thales walked the path.
      ::按下按钮查看Thales是如何走这条路的。

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    Similar or Not Similar?
    ::类似还是相似?

    By knowing that two of the corresponding angle pairs in the triangles were equal, Thales knew that the triangles would be similar. Thales knew this because he was familiar with the Angle-Angle Similarity Theorem . The Angle-Angle Similarity Theorem states that if two angles in one triangle are equal to two angles in another triangle, then the triangles are similar.
    ::Thales知道三角形中两个对应的角度对是相等的,知道三角形是相似的。Thales知道这一点,因为他熟悉角-角相似理论。角-角相似理论指出,如果一个三角形中两个角度等于另一个三角形中两个角度,那么三角形是相似的。

    Use the Angle-Angle Similarity Theorem to confirm that  the triangles in the interactive below are similar.
    ::使用角-角相似性定理来确认下方互动的三角形相似。

    INTERACTIVE
    Similar or Not Similar?
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    • Move the red points,  C  and  D .  
      ::移动红点C和D
    • Observe the angle measures. Are the triangles similar?  
      ::观察角度测量,三角形相似吗?
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    Finding  the Distance of the Ship
    ::寻找船舶的距离

    Thales knew that he had constructed similar triangles. Once the triangles were constructed, Thales used a proportion to compare  the sides of one triangle to the corresponding sides of the other triangle to find the distance of the ship from the shore.
    ::泰勒斯知道他曾建造过类似的三角形,一旦三角形建成,泰勒斯用一定比例的比值将一个三角形的侧面与另一个三角形的相应侧面进行比较,以寻找船舶与海岸的距离。

    Example
    ::示例示例示例示例

    How far was the ship from the shore in the picture below?
    ::船离海岸有多远?

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    AB is unknown, BC is labeled 60 feet, CD is labeled 45 feet, DE is labeled 330 feet
    ::AB未知,BC标签为60英尺,CD标签为45英尺,DE标签为330英尺

     

    A B B C = D E C D x 60 = 330 45 45 ⋅ x = 60 ⋅ 330 45 x = 19 , 800 x = 440

    ::ABBC=DECDx60=3304545*x60=3304545x19 800x=440

    The ship would be 440 feet from shore.
    ::离岸440英尺

    Use the interactive below to practice finding the distance of ships from the shore.  
    ::利用下面的交互操作 来寻找船只离海岸的距离

    INTERACTIVE
    Finding the Distance of Ships
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    • Drag the red points to adjust B ,  C , and D  . 
      ::拖曳红色点以调整 B、 C 和 D 。
    • Once you have created similar triangles, enter the distance between the ship and the shore in the box that appears using the measurements given in your calculations. Round to the nearest whole number.
      ::一旦您创建了相似的三角形,请在使用计算时给出的测量值显示的框中输入船舶与海岸之间的距离。圆到最近的整数。
    • Press the white button that appears to check your answer.
      ::按下检查您答案的白色按钮 。
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    Discussion Question
    ::讨论问题

    As you were answering the questions about the distance of the boat from the shore, did you notice any tricks that made it easier to solve for the distance ?
    ::当你回答关于船离海岸的距离的问题时 你有没有注意到 任何能让距离更容易解决的诡计?

     Summary
    ::摘要

    • Angle-Angle Similarity states that if two angles in a triangle are congruent to corresponding angles of another triangle, then the triangles are similar. The third angle is implied to be congruent by  the .
      ::角-角相似性表示,如果三角形中的两个角度与另一个三角形的相应角度相匹配,则三角形是相似的。第三个角度隐含着一个角度的相似性。
    • Similar triangles have proportional corresponding sides.
      ::类似的三角形有比例对应的边。