旋转对称

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Rotation Symmetry
::旋转对称

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A shape has symmetry if it can be indistinguishable from its transformed image.
::如果形状与其变形图像无法区分, 则形状具有对称性 。

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• A shape has rotation symmetry if there exists a rotation less than $\begin{array}{r}{360}^{\circ }\end{array}$ that carries the shape onto itself.
  ::形状具有旋转对称性, 如果旋转小于 360 , 则会将形状自动传送到自己身上 。
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• If you can rotate a shape less than $\begin{array}{r}{360}^{\circ }\end{array}$ about some point and the shape looks like it never moved, it has rotation symmetry.
  ::如果您可以旋转一个小于360的形状, 大约某个点, 而形状看起来好像从未移动过, 它有旋转对称 。

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There are 3 ways to name rotation symmetry.
::有三种方法可以命名旋转对称 。

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1. Order : The order of rotation symmetry of a geometric figure is the number of times you can rotate the geometric figure so that it looks exactly the same as the original figure before you get back to where you started. 
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   1. In  the lefthand image below, the shape has two positions that are indistinguishable , so it has rotation symmetry of order 2 .
      ::在下面的左手图像中,形状有两个无法区分的方位,因此它具有顺序2的旋转对称性。
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   2. On the right, the shape has the positions that are indistinguishable , so it has rotation symmetry of order 3 .
      ::在右侧,形状的姿势是无法区分的, 因此它具有顺序3的旋转对称性。
   
   ::顺序 : 几何图的旋转对称顺序是您可以旋转几何图的次数, 以使它看起来与您回到您开始的位置之前的原始图形完全相同。 在下面的左手图像中, 形状有两个无法区分的位置, 因此它具有顺序2 的旋转对称性 。 在右边, 形状有无法区分的位置, 所以它具有顺序3 的旋转对称性 。

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2. Fraction of a turn:  If you look at how far you turn the shape to get it to look the same, you can think of that as a fraction of how far you go to get all the way around.
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   1. On the left, you turn $\begin{array}{r}\frac{1}{2}\end{array}$ way around, so it has $\begin{array}{r}\frac{1}{2}\end{array}$ turn symmetry.
      ::在左边,你转过12圈, 所以它有12转对称。
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   2. On the right, it turns $\begin{array}{r}\frac{1}{3}\end{array}$ of the way around to look like itself again, so it has $\begin{array}{r}\frac{1}{3}\end{array}$ turn symmetry.
      ::在右边,它转过13个方向 重新看起来像自己, 所以它有13个方向对称。
   
   ::转弯的分数 : 如果您查看形状的变形有多远才能让形状看起来一样, 您可以将它视为你走过的路程的一小部分。 在左边, 您转过12公里, 因此它有12个对称。 在右边, 它转过13个方向再看一次, 所以它有13个对称 。

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3. Angle : The angle of rotation symmetry is the smallest angle the figure can be rotated to coincide with itself. So the half turn becomes half of $\begin{array}{r}{360}^{\circ }={180}^{\circ },\end{array}$  and the one-third turn becomes a third of $\begin{array}{r}{360}^{\circ }={120}^{\circ }.\end{array}$
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   1. The shape on the left has $\begin{array}{r}{180}^{\circ }\end{array}$ rotation symmetry.
      ::左侧的形状有180 旋转对称。
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   2. T he shape on the right has $\begin{array}{r}{120}^{\circ }\end{array}$ rotation symmetry.
      ::右侧的形状有 120 旋转对称 。
   
   ::角度 : 旋转对称角度是该图可以旋转以与其同步的最小角度 。 因此, 半转为 360 180 的半转, 三分之一转为 360 120 的三分之一 。 左边的形状是 180 旋转对称。 右边的形状是 120 旋转对称 。

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Identifying Rotation Symmetry
::识别旋转对称

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A rectangle is an example of a shape with rotation symmetry .
::矩形是旋转对称形状的示例。

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• A rectangle can be rotated $\begin{array}{r}{180}^{\circ }\end{array}$ about its center and it will look exactly the same and be in the same location. The only difference is the location of the named points.
  ::矩形可以旋转 180 围绕其中心, 它会看起来完全一样, 并且在同一位置。 唯一的区别是指定点的位置 。
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• A rectangle has half-turn symmetry, and therefore is order 2.
  ::矩形有半转对称,因此是顺序2。

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Does a square have rotation symmetry ?
::方形有旋转对称吗?

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Yes, a square can be rotated $\begin{array}{r}{90}^{\circ }\end{array}$ counterclockwise (or clockwise) about its center and the image will be indistinguishable from the original square.
::是的, 一个正方形的中间可以旋转 90 / countplack why( 或顺时针) , 图像将与原始正方形无法区分 。

 

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Identifying Angles of Rotation
::识别旋转角度

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How many angles of rotation cause a square to be carried onto itself?
::有多少旋转角度导致一个方形被自动携带到自己身上?

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• Rotations of $\begin{array}{r}{90}^{\circ },{180}^{\circ },\end{array}$   and $\begin{array}{r}{270}^{\circ }\end{array}$  in either direction  will all cause the square to be carried onto itself.
  ::任何方向的90,180,和270的旋转 都会导致广场自行移动
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• A square has quarter-turn rotation symmetry, and so  has an order of 4.
  ::方形有四分之一的旋转对称,四分左右的旋转对称。

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Rotation Symmetry of a Trapezoid
::轨迹的旋转对称

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Consider  rotation symmetry in .
::考虑旋转对称 。

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• In a generic trapezoid as well as isosceles trapezoid , there is no rotation symmetry.
  ::在一般的捕鲸类和等离子类捕鲸类中,没有旋转对称。
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• A trapezoid must be rotated a full $\begin{array}{r}{360}^{\circ }\end{array}$ to again appear in its original position.
  ::陷阱类体必须旋转 360 才能再次出现在它原来的位置上

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Rotation Symmetry of a Circle
::圆圆的旋转对称

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R otate  a circle about its center $\begin{array}{r}O\end{array}$ through any angle  and it fits onto itself.
::旋转一个环绕其中心O的圆,通过任何角度旋转,并适合它本身。

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• A circle has rotation symmetry around the center for every angle.
  ::每个角度的中心周围都有旋转对称的圆。
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• A circle has an unlimited number of angles of symmetry and the order of its rotation is infinite.
  ::一个圆的对称角度数是无限的,其旋转顺序是无限的。

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Rotation Symmetry
::旋转对称

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Drag the cursor to rotate the polygon and observe the angle of rotation.
::拖曳光标以旋转多边形并观察旋转角度。

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Examples
::实例实例实例实例

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Example 1
::例1

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What happens when you rotate the regular pentagon below $\begin{array}{r}{72}^{\circ }\end{array}$ clockwise about its center? Why is $\begin{array}{r}{72}^{\circ }\end{array}$ special in this case?
::当您将常规的五角形旋转到 72 度以下时钟方向中心时会怎么样?为什么在这种情况下,72 度是特别的呢?

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When you rotate the regular pentagon $\begin{array}{r}{72}^{\circ }\end{array}$ about its center, it will look exactly the same. This is because the regular pentagon has rotation symmetry, and $\begin{array}{r}{72}^{\circ }\end{array}$ is the minimum number of degrees you can rotate the pentagon in order to carry it onto itself.
::当您旋转常规的五角形 72 左右其中心时, 它会看起来完全一样。 这是因为普通的五角形具有旋转对称性, 而 72 是您可以旋转的五角形最低度数, 以便将它自动移到它身上 。

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Example 2
::例2

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Does  each capital letter below have rotation symmetry? If so, state the angles of rotation that carry the letter onto itself.
::下面的每个大写字母是否具有旋转对称性?如果是,请说明将字母自动移动的旋转角度。

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1. Capital letter N 
   ::首字母N

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Yes, it does have rotation symmetry . It can be rotated $\begin{array}{r}{180}^{\circ }.\end{array}$
::是的,它有旋转对称,可以旋转180

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2. Capital l etter S 
   ::英文大写字母 S

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Yes, it does have rotation symmetry . It can be rotated $\begin{array}{r}{180}^{\circ }.\end{array}$
::是的,它有旋转对称,可以旋转180

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3. Capital letter H
   ::英文大写字母H

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Yes, it does have rotation symmetry . It can be rotated $\begin{array}{r}{180}^{\circ }.\end{array}$
::是的,它有旋转对称,可以旋转180

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4. Capital letter B
   ::英文大写字母B

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No, it  does not have rotation symmetry.
::不,它没有旋转对称。

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CK-12 PLIX Interactive
::CK-12 PLIX 交互式互动

Summary

播放段落 Rotation symmetry occurs when a shape can be rotated less than 360° and still look the same. ::当形状旋转小于360°且看起来仍然相同时,就会发生旋转对称。 播放段落 There are three ways to name rotation symmetry: order, fraction of a turn, and angle. ::有三种方法可以命名旋转对称:顺序、转角的分数和角度。 播放段落 A circle has infinite rotation symmetry, as it can be rotated around its center through any angle and still fit onto itself. ::圆具有无限的旋转对称性,因为它可以通过任何角度在圆中心周围旋转,并且仍然适合自己。

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1. What does it mean for a shape to have symmetry?
::1. 形状对称意味着什么?

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2. What does it mean for a shape to have rotation symmetry?
::2. 形状的旋转对称意味着什么?

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3. Why does the stipulation of less than $\begin{array}{r}{360}^{\circ }\end{array}$  exist in the definition of rotation symmetry?
::3. 为什么轮换对称定义中存在低于360的规定?

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For each of the following shapes, state whether or not it has rotation symmetry. If it does, state the number of degrees you can rotate the shape to carry it onto itself.
::对于以下每个形状, 请说明它是否有旋转对称。 如果有, 请说明您可以旋转形状的度数, 以将形状自动移动 。

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4. Equilateral triangle
::4. 等边三角形

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5. Isosceles triangle
::5. 悬浮三角形

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6. Scalene triangle
::6. 缩缩三角形

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7. Parallelogram
::7. 平行图

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8. Rhombus
::8. 滚轮

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9. Regular pentagon
::9. 经常五边形

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10. Regular hexagon
::10. 普通六边形

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11. Regular 12-gon
::11. 经常12个角

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12. Regular $\begin{array}{r}n\end{array}$ -gon
::12. 经常正正正

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13. Circle
::13. 圆环

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14. Kite
::14. 基特语

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15. Where will the center of rotation always be located for shapes with rotation symmetry?
::15. 在旋转对称形状方面,旋转中心总是在哪里?

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16. Does every polygon that has rotation symmetry also have reflection symmetry? Why or why not?
::16. 每个具有旋转对称性的多边形是否也具有反射对称性?为什么或为什么没有?

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17. Does every polygon that has reflection symmetry also have rotation symmetry? Why or why not?
::17. 每个反射对称的多边形是否也具有旋转对称性?为什么或为什么没有?

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18. Does a line have rotation symmetry? How about reflection symmetry? Explain.
::18. 线条是否具有旋转对称性?反射对称性如何?解释一下。

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19. Does an angle have rotation symmetry? How about reflections symmetry? Explain?
::19. 角度是否具有旋转对称性?反射对称性如何?解释?

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20. Harold argues that a semi-circle has rotation symmetry, but Jay disagrees. Who is correct and why?
::20. Harold认为,半圆环具有轮换对称性,但Jay不同意。

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21. Tina argues that our reflections in the mirror show that humans have rotational symmetry, the image in a mirror is a $\begin{array}{r}{180}^{\circ }\end{array}$ rotation of the original. Brenda disagrees. Who is correct and why?
::21. 蒂娜认为,我们在镜子中的反射表明,人类的轮回对称,镜子中的图像是原版180的旋转。布伦达不同意,谁是正确的,为什么?

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22. Vanessa wants a house with a floor plan that has reflection symmetry, while Jessica prefers one that has rotation symmetry. Draw sketches of each. Which do you prefer and why?
::22. Vanessa想要一栋有平面平面平面图的房子,而Jessica更喜欢有旋转对称的房屋,绘制每个房屋的草图。

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23. Many animals, plants, or other objects in the world have reflection or rotation symmetry. Give examples and describe the symmetry in as much detail as possible.
::23. 世界上许多动物、植物或其他物体具有反射或旋转对称性,请举例说明并尽可能详细地描述对称性。

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Review (Answers)
::审查(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 。