1.5 确定共性-interactive

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  Congruence in Animation
  ::动动中的一致性

  

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  Previously, you've explored rigid motion transformations in animation and graphic design. You learned that when a shape undergoes a translation, reflection or rotation , the side lengths and angles of the image and the pre-image do not change.
  ::之前,您曾探索过动画和图形设计中的僵硬动作转换。 您学会了当一个形状被翻译、 反射或旋转时, 图像的侧边长度和角度以及图像预览不会改变 。

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  Images or shapes that are the same shape and size are called  congruent .  If two shapes are congruent, their  corresponding sides are the same length and their are equal. 
  ::相同形状和大小的图像或形状被称为相容。 如果两个形状是相容的, 它们对应的两面长度相同, 它们相等 。

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  It is important in animation that a shape that has undergone a rigid motion transformation is congruent with its pre-image. The size of a character on a screen  sends information to your brain about where the character is in the environment. Animators can increase or decrease a character's size on purpose to create a particular effect, but all animators must know how different transformations affect the size and shape of a character or object.
  ::在动画中,必须使经历僵硬运动变形的形状与其预图象一致。屏幕上一个字符的大小会向大脑传递关于该字符在环境中的位置的信息。动画家可以有意增加或减少一个字符的大小以产生特定效果,但所有动画家都必须知道不同的变形会如何影响一个字符或对象的大小和形状。

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  Using the interactive below, match any shapes  that are congruent. Then answer the questions below it.
  ::使用下面的交互效果, 匹配任何相容的形状。 然后回答下面的问题 。

   

   

   

   

   

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  Mapping S hapes
  ::绘图形状

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  When you dragged the shapes (above) to help determine if they were congruent, you were performing what is called shape mapping . Shape mapping is the use of transformations to move one shape onto another. When  you use these transformations to match up corresponding sides and angles, you can often see if the two shapes are congruent.
  ::当拖动形状( 以上) 以帮助确定它们是否相容时, 您正在执行所谓的形状映射。 形状映射是使用变形将一个形状移动到另一个形状。 当您使用这些变形来匹配相应的侧面和角度时, 您经常可以看到这两个形状是否相容 。

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  Use the interactive below to map one shape onto another using either a translation, reflection, or rotation, to determine whether or not the shapes are congruent.  For an extra challenge ,  see if you can line up each shape  using the minimum number of transformations.
  ::使用下面的交互效果将一个形状映射成另一个形状, 使用翻译、 反射或旋转来决定形状是否相容。 对于额外的挑战, 请查看您是否可以使用最小的变换数排列每个形状 。

   

   

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  Discussion Question
  ::讨论问题

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  What were some of the challenges while shape mapping?  Generalize an approach for mapping the shape in the fewest steps possible.
  ::绘制形状时遇到的一些挑战是什么?以尽可能少的步骤概括绘制形状的方法。

   

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  Mapping Lines
  ::绘图线

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  Take lines to lines through rotations, reflections , and translations in the interactive below.
  ::通过旋转、反射和翻译到线条,在下文互动中进行线条。

   

   

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  Discussion Question
  ::讨论问题

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  What were some of the challenges while mapping the lines? How did it differ compared to mapping shapes? Did you run into similar challenges? Did you use any of the strategies you used when mapping shapes?
  ::绘制线条时遇到哪些挑战? 与绘制形状相比,它有什么不同? 您遇到过类似的挑战吗? 您在绘制形状时使用过任何策略吗?

   

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  Measuring C ongruence
  ::衡量一致性

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  One way to check whether two shapes are congruent is to overlap them and compare their shapes and sizes. However,  when checking for congruence, it  often isn't that easy. The best way to check for congruence is to measure the sides and angles of each shape. This is a practice that is extremely important in engineering and construction. A wide  variety of objects in our society are made using  congruent shapes. 
  ::检查两种形状是否一致的一个方法就是将两者重叠并比较其形状和大小。 但是, 当检查是否一致时, 它往往不是那么容易。 检查一致性的最佳方法就是测量每个形状的侧面和角度。 这是工程和构造中极为重要的一种做法。 我们社会上各种各样的物体都是用一致的形状制作的。

  

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  When one of these structures or objects is designed, even the slightest error in angles or side lengths could hurt its integrity. To account for this, sides and angles are meticulously measured to ensure congruence.
  ::当这些结构或物体之一被设计时,即使角度或侧边的微小差错也会损害其完整性。 为了说明这一点,对边和角度进行了仔细的测量,以确保一致性。

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  In the interactive below, use the ruler tool and protractor tool to determine if the two shapes are congruent.
  ::在下文互动部分,使用标尺工具和减号工具确定两种形状是否一致。

   

   

   

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  Congruence Statements
  ::一致发言

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  When two shapes are congruent,  you can write a congruence statement to indicate which sides and angles are congruent. The symbol used to show congruence is  $\begin{align*}\cong.\end{align*}$  
  ::当两个形状一致时,您可以写一个一致的语句,以表示哪些侧面和角度是一致的。用于显示一致性的符号是 \\\ \ \ 。

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  Example:  If a statement says  $\begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}$  (read as "triangle  ABC  is congruent to triangle  XYZ" ),  it specifies which vertices correspond. Vertex  $\begin{align*}A\end{align*}$  corresponds to  $\begin{align*}X,\end{align*}$  vertex  $\begin{align*}B\end{align*}$  corresponds to  $\begin{align*}Y,\end{align*}$  and vertex  $\begin{align*}C\end{align*}$  corresponds to  $\begin{align*}Z.\end{align*}$
  
  ::示例:如果一个语句表示 ABC XYZ( 读作“ 三角ABC 与三角 XYZ 相同 ” ) , 它会指定对应的顶部。 Vertex A 对应 X, 顶部 B 对应 Y , 顶部 C 对应 Z 。

  

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  Corresponding angles and sides of congruent shapes will always be congruent.  You can use a congruence statement to show that sides or angles are congruent.
  ::对应角度和相近形状的侧面总是一致的。 您可以使用一致的语句来显示侧面或角度是一致的 。

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  • The statement  $\begin{align*}\overline{AB} \cong \overline{XY}\end{align*}$   (read as "side AB  is congruent to side XY ")  states that side  $\begin{align*}AB\end{align*}$  is the same length as side  $\begin{align*}XY.\end{align*}$  
    ::声明 AB {XY } ( 读为“ AB 侧与 XY 侧一致”) 表示, AB 侧与 XY 侧的长度相同 。
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  • The statement $\begin{align*}\angle BCA \cong \angle YZX\end{align*}$ (read as "angle BCA  is congruent to angle YZX") states that angle  $\begin{align*}BCA \end{align*}$  is the same measure as angle $\begin{align*}YZX.\end{align*}$
    ::BCAYZX( 读为“ 角 BCA 与角 YZX 相同) 的语句中写道, 角 BCA 与角 YZX 相同 。

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  Use the interactive below to explore this further and answer the associated questions.
  ::用下面的交互方式进一步探讨这个问题,并回答相关问题。

   

   

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    Summary
  ::摘要

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  • Images or shapes that are exactly the same shape and size are called congruent.
    ::与形状和大小完全相同的图像或形状被称为相容的图像或形状。
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  • Corresponding angles and sides of congruent shapes will always be congruent. This fact is sometimes  referred to  as "Corresponding Parts of Congruent Triangles are Congruent," and abbreviated as "CPCTC."
    ::相应的角度和相近形状的侧面总是一致的。 这一事实有时被称为“ 相匹配的康格伦三角系部分是康格伦特 ” , 缩写为“ CPCTC ” 。
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  • The symbol used to show congruence is  $\begin{align*}\large{\cong}.\end{align*}$
    ::用于显示一致性的符号是 。