2.6 反射对称-interactive

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  反射对称

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  Reflection Symmetry
  ::反射对称

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  • A shape has symmetry if it is 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 reflection symmetry if there  is a line through the center of the shape that you can reflect across without the shape appearing to move at all.
    ::形状有反射对称, 如果在形状的中心有一条线, 您可以反射到形状的中间, 而形状却完全没有移动 。
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  • This line of reflection is called a line of symmetry .
    ::这种反射线被称为对称线。

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  In other words, a line of symmetry is a line that divides a figure into two mirror images. The figure is mapped onto itself by a reflection in this line. Some figures have one or more , while other figures have no lines of symmetry.
  ::换句话说,对称线是将一个图分为两个镜像图像的线条。图是用反射线绘制的。有些图有一个或多个,而其他图则没有对称线。

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  A rectangle is an example of a shape with reflection symmetry. A line of reflection through the midpoints of opposite sides will always be a line of symmetry.
  ::矩形是反射对称形状的例子。 反射线穿过对面两侧中点的反射线总是对称线。

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  A rectangle has two lines of symmetry . You can imagine folding the rectangle along each line of symmetry and each half of the rectangle would match up perfectly. Remember that a shape has to have at least one line of symmetry for it to be considered a shape with reflection symmetry .
  ::矩形有两条对称线。 您可以想象每条对称线折叠矩形, 每条对称线每半个矩形都会完全匹配。 记住, 形状必须至少有一条对称线才能被视为反射对称的形状 。

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  Recognizing Reflection Symmetry
  ::确认反反射对称

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  Let's consider a regular hexagon as an example.
  ::让我们以普通的六边形为例

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  • A regular hexagon has all its sides congruent and each of the angles measure $\begin{array}{r}{120}^{\circ }.\end{array}$
    ::普通六边形的两面均匀,每个角度的度量为120。
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  • A regular hexagon has a number of lines of reflection: three along the lines joining the midpoints of its opposite sides and three along the diagonals .
    ::正常的六边形有几条反射线:三条线与对面两侧中点交接,三条线与对角线交接。
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  • Thus, a hexagon has reflection symmetry .
    ::因此,六边形具有反射对称性。

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  Identifying Lines of Symmetry
  ::识别对称线

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  A square is an example of a shape with reflection symmetry.
  ::方形是反射对称形状的一个示例。

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  • In a square, all sides are congruent and each angle is a right angle .
    ::在方形中,各方是一致的,每个角度都是正确的角度。
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  • There are  four lines of reflection that carry the square onto itself. These lines of reflection will always be the lines of symmetry .
    ::这些反射线总是对称线。
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  • Two lines of symmetry separate the square into two equal , whereas the other two separate the square into two equal triangles.
    ::两条对称线将方形分为两对等,而另外两条则将方形分为两对等三角形。

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  Identify the line of reflection symmetry in the given figure.
  ::在给定数字中标明反射线对称。

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  is the line of reflection symmetry.
  ::是反射对称线。

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  Reflection Symmetry in Trapezoids
  ::轨迹的反射对称

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  A generic trapezoid will not have reflection symmetry.
  ::普通的捕鲸类不会有反射对称 。

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  • An isosceles trapezoid will have reflection symmetry because the line connecting the midpoints of the bases will be a line of symmetry .
    ::等离子捕鲸类将具有反射对称性,因为连接基地中点的线将是一个对称线。

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  Reflection Symmetry
  ::反射对称

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  Explore the reflection symmetry of the selected shape. Click the blue arrow to activate the figure, then drag the cursor to draw as many lines of symmetry as are given for each shape.
  ::探索所选形状的反射对称。 单击蓝箭头来激活图形, 然后拖动光标来绘制每个形状的对称行数。

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  • Rectangle: 2
    ::矩形: 2
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  • Triangle : 3
    ::三角形: 3
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  • Pentagon : 5
    ::五角角: 5
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  • Circle : ? How many can you find?
    ::圆:你能找到多少个?

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  Triangle
  ::三角三角形

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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 reflect the regular pentagon below across line $\begin{array}{r}f?\end{array}$  Why is the line of reflection in this case called a line of symmetry?
  ::当你反射到F线下方的正五角形时会发生什么?为什么反射线在这个案例中被称为对称线?

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  When you reflect the regular pentagon below across line $\begin{array}{r}f,\end{array}$  the pentagon will look exactly the same. Without labeled points, it will be impossible to tell the difference between the original pentagon and its image. This means that the pentagon has reflection symmetry. Line $\begin{array}{r}f\end{array}$ is a line of symmetry because it's the line across which the pentagon can be reflected without visible change.
  ::当您反射光线 f 下方的正五边形时, 五边形将看起来完全相同。 没有标签点, 就无法辨别原始的五边形与其图像的区别。 这意味着五边形反射对称。 线 F 是一条对称线, 因为五边形的反射线是一条不可见变化的线。

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  Example 2
  ::例2

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  Do the capital letters below have reflection symmetry? If so, state how many and where these lines of symmetry are.
  ::下面的大写字母是否有反射对称性?如果有,请说明这些对称性行的数目和位置。

  1.  

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  T he capital C has a line of reflection going horizontally through its center, and it looks the same on the top and the bottom, so we say  that C has a horizontal line of symmetry.
  ::首都C有一个反射线 横向穿透中心, 它在顶部和底部看起来是一样的, 所以我们说,查斯的横向对称线。

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  The capital M has a line of reflection going through its center where the M is the same on the left and right. Because the line of reflection is vertical, we say M has a vertical line of symmetry.
  ::首都M 的反射线穿过中心,中间的M在左右是相同的。 因为反射线是垂直的,我们说M有垂直的对称线。

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  T he H has two lines of reflection - the vertical line through its center and the horizontal line through its center are both lines of reflection, so H has both a horizontal and a vertical line of symmetry.
  ::H 有两条反射线- 穿过它的中心垂直线和穿过它的中心水平线都是反射线, 所以H有一条水平对称线和一条垂直对称线。

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  No, the capital F does not have any line of reflection - you can't split it into two halves that are exactly the same on both sides of a line.
  ::不,首都F没有反射线 你不能把它分成两半,两半 线两侧完全一样。

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  CK-12 PLIX Reflection Symmetry
  ::CK-12 PLIX 反射对称

  Summary

  播放段落 Reflection symmetry occurs when a shape is indistinguishable from its transformed image after reflecting across a line through its center. ::反射对称发生于一个形状在通过中心反射过一条线后,与其变形图像无法区分时。 播放段落 The line of symmetry is this line of reflection that divides a figure into two mirror images. ::对称线是将数字分为两个镜像图像的反射线。

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  Review
  ::审查审查审查审查

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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 reflection symmetry?
  ::2. 形状的反射对称意味着什么?

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  3. What do you think it would mean for a shape to have translation symmetry? Can you think of any shapes or objects with translation symmetry?
  ::3. 你认为一个形状具有翻译对称意味着什么?你能想到任何带有翻译对称的形状或物体吗?

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  For each of the following shapes, state whether or not it has reflection symmetry. If it does, state how many lines of symmetry it has and describe where the lines of symmetry are.
  ::对于以下的形状中的每一形状,请说明它是否有反射对称。如果有,请说明它有多少对称线,并描述对称线的位置。

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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. In order to have reflection symmetry, must a polygon have at least two sides that are the same length? Explain.
  ::15. 为了具有反射对称性,多边形必须至少有两面长度相同的两面吗?

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  16. Give examples of objects with reflection symmetry in nature.
  ::16. 举例说明反射对称性质的物体。

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  17. Does every regular polygon have at least one line of symmetry? How many does an equilateral triangle have? A square? A pentagon? Experiment. (In interactive geometry software, in the dropdown for polygon creation, one can create a regular polygon of any number of desired sides. One can also create a perpendicular bisector of a segment, and an angle bisector.) How does the number of lines of symmetry change as the number of sides on the regular polygon changes? Why? How do the characteristics of these lines of symmetry change as the number of sides increase? Why?
  ::17. 每个正态多边形至少有一对称线吗?一个正方形三角有多少对称线?一个正方形?一个五边形?实验。 (在交互式几何软件中,在多边形创建的下调中,人们可以创造出任何几个理想边的正多边形,也可以创造出一个分形的直角双形和角两边的对称线。 )当正态多边形的边数发生变化时,对称线的变化数量如何?为什么?这些对称线随着边数的增加而变化的特点如何? 为什么?

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  18. Construct a line, then three non-collinear points on one side.  Then reflect the points across a line. Connect these points to form a polygon which has reflection symmetry. Use this construction to explain the meaning of reflection symmetry and the characteristics of objects which have it.
  ::18. 构造一条线, 然后在一面构造三个非对线点, 然后反射横线的点。 将这些点连接到一个具有反射对称的多边形上。 使用此构造来解释反射对称的含义和有反射对称的物体的特性 。

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  19. If a figure with reflection symmetry is reflected across a line that does not   intersect it, is there a translation and rotation that will map the image to the original? Experiment and discuss. Is this different from polygons that do not have reflection symmetry? Why or why not?
  ::19. 如果反射对称图在不相交的线上得到反映,是否有翻译和旋转将图像映射为原始图像?实验和讨论,这是否不同于没有反射对称的多边形?为什么或为什么没有?

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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 。