12.4 轮调

Section outline

• Select section General

  Collapse Expand

  General

  Collapse all Expand all

  • Select activity 新闻通告

    

    新闻通告 Forum
• Select section 轮调

  Collapse Expand

  轮调

  播放段落

  Rotations
  ::轮调

  播放段落

  A transformation is an operation that moves, flips, or otherwise changes a figure to create a new figure. A rigid transformation (also known as an isometry or congruence transformation ) is a transformation that does not change the size or shape of a figure.
  ::变换是一种移动、翻转或以其他方式改变数字以创建新图的操作。 僵硬变换( 也称为等离线或相容变换) 是一种不会改变图的大小或形状的变换 。

  播放段落

  The rigid transformations are translations, reflections , and rotations . The new figure created by a transformation is called the image . The original figure is called the preimage . If the preimage is $\begin{align*}A\end{align*}$ , then the image would be $\begin{align*}A'\end{align*}$ , said “a prime.” If there is an image of $\begin{align*}A'\end{align*}$ , that would be labeled $\begin{align*}A''\end{align*}$ , said “a double prime.”
  ::硬质变换是翻译、反射和旋转。 由变换创造的新数字被称为图像。 原始数字被称为预映。 如果预映为A,那么图像将是A , “ 黄金 ” 。 如果有A 的图像,则标为A 的A , “双质 ” , 则表示“双质 ” 。

  播放段落

  A rotation is a transformation where a figure is turned around a fixed point to create an image. The lines drawn from the preimage to the center of rotation and from the center of rotation to the image form the angle of rotation .
  ::旋转是一种转换, 将数字围绕固定点旋转, 以创建图像。 从预映到旋转中心, 从旋转中心到图像中心的线条构成旋转角度 。

  

  播放段落

  While we can rotate any image any amount of degrees, $\begin{align*}90^\circ, 180^\circ\end{align*}$ and $\begin{align*}270^\circ\end{align*}$ rotations are common and have rules worth memorizing.
  ::当我们可以旋转任何水平的图像时, 90°, 180°, 270° 的旋转是常见的, 并且有值得记住的规则 。

  播放段落

  Rotation of $\begin{align*}180^\circ\end{align*}$ : $\begin{align*}(x,y) \rightarrow (-x,-y)\end{align*}$
  ::180(x,y)(-x,-y)的旋转

  播放段落

  Rotation of $\begin{align*}90^\circ\end{align*}$ : $\begin{align*}(x,y) \rightarrow (-y,x)\end{align*}$
  ::90(x,y)(-y,x)的旋转

  播放段落

  Rotation of $\begin{align*}270^\circ\end{align*}$ : $\begin{align*}(x,y) \rightarrow (y,-x)\end{align*}$
  ::270(x,y)(y,-x)的旋转

  播放段落

  What if you were given the coordinates of a quadrilateral and you were asked to rotate that quadrilateral $\begin{align*}270^\circ\end{align*}$ about the origin ? What would its new coordinates be?
  ::如果四边形的座标给了你 并且要求你旋转原位的四边形270?

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  A rotation of $\begin{align*}80^\circ\end{align*}$ clockwise is the same as what counterclockwise rotation?
  ::逆时针旋转 80 的顺时针旋转与逆时针旋转相同吗 ?

  播放段落

  There are $\begin{align*}360^\circ\end{align*}$ around a point. So, an $\begin{align*}80^\circ\end{align*}$ rotation clockwise is the same as a $\begin{align*}360^\circ-80^\circ=280^\circ\end{align*}$ rotation counterclockwise.
  ::环绕一个点有 360 。 所以, 80 顺时针旋转与 360 80 280 逆时针旋转相同 。

  

  播放段落

  Example 2
  ::例2

  播放段落

  A rotation of $\begin{align*}160^\circ\end{align*}$ counterclockwise is the same as what clockwise rotation?
  ::逆时针 160 的旋转与顺时针的旋转相同吗?

  播放段落

  $\begin{align*}360^\circ-160^\circ=200^\circ\end{align*}$ clockwise rotation.
  ::360 160

  

  播放段落

  Example 3
  ::例3

  播放段落

  Rotate $\begin{align*}\overline{ST} \ 90^\circ\end{align*}$ .
  ::零下90 。

  

  

  播放段落

  Example 4
  ::例4

  播放段落

  The rotation of a quadrilateral is shown below. What is the measure of $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ ?
  ::以下显示四边形的旋转。 x 和 y 的量度是多少?

  

  播放段落

  Because a rotation produces congruent figures, we can set up two equations to solve for $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ .
  ::因为旋转产生一致的数字, 我们可以为 x 和 y 设置两个方程式解析 。

  播放段落

  $$\begin{align*}2y &= 80^\circ && 2x-3=15\\ y &= 40^\circ && \quad \ \ 2x=18\\ & && \qquad \ x = 9\end{align*}$$
  ::2y=802x-3=15y=402x=18 x=9

  播放段落

  Example 5
  ::例5

  播放段落

  Rotate $\begin{align*}\triangle ABC\end{align*}$ , with vertices $\begin{align*}A(7, 4), B(6, 1)\end{align*}$ , and $\begin{align*}C(3, 1)\end{align*}$ , $\begin{align*}180^\circ\end{align*}$ about the origin. Find the coordinates of $\begin{align*}\triangle A'B'C'\end{align*}$ .
  ::旋转 ABC, 上面有 A( 7) 4, B( 6, 1) 和 C( 3, 1), 180 有关源。 找到 {A_ B_ C} 的坐标 。

  

  播放段落

  Use the rule above to find $\begin{align*}\triangle A'B'C'\end{align*}$ .
  ::使用上述规则来找到______________________________________________________B________________________________________________________________________________________________________________________________________

  播放段落

  $$\begin{align*}A(7,4) & \rightarrow A'(-7,-4)\\ B(6,1) & \rightarrow B'(-6,-1)\\ C(3,1) & \rightarrow C'(-3,-1)\end{align*}$$
  ::A(7,4)A*(-7,7)B(6,1)B(6)B(6)C(3,1)C(3,1)C[3,3)C[3,-1)B(6,1)B[6)-(6)C(3,1)C__(3,3)-(1)

  播放段落

  Review
  ::回顾

  播放段落

  In the questions below, every rotation is counterclockwise , unless otherwise stated.
  ::在下面的问题中,除另有说明外,每次轮调都是逆时针。

  播放段落
  1. If you rotated the letter $\begin{align*}p \ 180^\circ\end{align*}$ counterclockwise, what letter would you have?
     ::如果你逆时针旋转字母p180, 你会有什么字母?
  播放段落
  2. If you rotated the letter $\begin{align*}p \ 180^\circ\end{align*}$ clockwise , what letter would you have?
     ::如果你按时间方向旋转字母 p180, 你会有什么字母?
  播放段落
  3. A $\begin{align*}90^\circ\end{align*}$ clockwise rotation is the same as what counterclockwise rotation?
     ::90 顺时针旋转与什么逆时针旋转相同?
  播放段落
  4. A $\begin{align*}270^\circ\end{align*}$ clockwise rotation is the same as what counterclockwise rotation?
     ::270 顺时针旋转与逆时针旋转相同 ?
  播放段落
  5. A $\begin{align*}210^\circ\end{align*}$ counterclockwise rotation is the same as what clockwise rotation?
     ::逆时针旋转与按时针旋转相同?
  播放段落
  6. A $\begin{align*}120^\circ\end{align*}$ counterclockwise rotation is the same as what clockwise rotation?
     ::120 逆时针旋转与按时针旋转相同?
  播放段落
  7. A $\begin{align*}340^\circ\end{align*}$ counterclockwise rotation is the same as what clockwise rotation?
     ::逆时针旋转与按时针旋转相同?
  播放段落
  8. Rotating a figure $\begin{align*}360^\circ\end{align*}$ is the same as what other rotation?
     ::数字的旋转360 * 与其它的旋转相同吗?
  播放段落
  9. Does it matter if you rotate a figure $\begin{align*}180^\circ\end{align*}$ clockwise or counterclockwise? Why or why not?
     ::旋转数字180时钟还是反时钟有关系吗?为什么或为什么没有?
  播放段落
  10. When drawing a rotated figure and using your protractor, would it be easier to rotate the figure $\begin{align*}300^\circ\end{align*}$ counterclockwise or $\begin{align*}60^\circ\end{align*}$ clockwise? Explain your reasoning.
      ::当绘制旋转的图形并使用您的减速器时, 比较容易旋转数字 300 或 六十 或 时钟 。 请解释您的推理 。

  播放段落

  Rotate each figure counterclockwise in the coordinate plane the given angle measure. The center of rotation is the origin.
  ::在坐标平面中逆时针旋转每个图,给定角度量度。旋转的中心是源。

  11. $\begin{align*}180^\circ\end{align*}$
  12. $\begin{align*}90^\circ\end{align*}$
  13. $\begin{align*}180^\circ\end{align*}$
  14. $\begin{align*}270^\circ\end{align*}$
  15. $\begin{align*}90^\circ\end{align*}$
  16. $\begin{align*}270^\circ\end{align*}$
  17. $\begin{align*}180^\circ\end{align*}$
  18. $\begin{align*}270^\circ\end{align*}$
  19. $\begin{align*}90^\circ\end{align*}$

  播放段落

  Find the measure of $\begin{align*}x\end{align*}$ in the rotations below. The blue figure is the preimage.
  ::在下面的旋转中查找 x 的度量。 蓝色数字是预视图。

  20. 
  21. 
  22. 

  播放段落

  Find the counterclockwise angle of rotation for the graphs below. The center of rotation is the origin and the blue figure is the preimage. Your answer will be  $\begin{align*}90^\circ, 270^\circ\end{align*}$ , or $\begin{align*}180^\circ\end{align*}$ .
  ::查找下图的逆时针旋转角度。 旋转的中心是来源, 蓝色是预映。 您的回答是 90 , 270 , 或 180 。

  23. 
  24. 
  25. 
  26. 
  27. 
  28. 

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  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 。