课程： 12.3 几何翻译 | The Way To Learn

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选择节几何翻译

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几何翻译
Translations
::笔译翻译

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。

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 $\begin{aligned} A \end{aligned}$ , then the image would be $\begin{aligned} A^{'} \end{aligned}$ , said “a prime.” If there is an image of $\begin{aligned} A^{'} \end{aligned}$ , that would be labeled $\begin{aligned} A^{″} \end{aligned}$ , 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 ” 。

A translation is a transformation that moves every point in a figure the same distance in the same direction. For example, this transformation moves the to the right 5 units and up 3 units. It is written $\begin{aligned} (x, y) \to (x + 5, y + 3) \end{aligned}$ .
::翻译是一种转换, 将数字中的每个点都移动在同一方向的相同距离。例如, 此转换会向右移到 5 个单位, 向上移到 3 个单位。它会写入 (x, y) {( x+5,y+3) 。

What if you were given the coordinates of a quadrilateral and you were asked to move that quadrilateral 3 units to the left and 2 units down? What would its new coordinates be?
::如果你们获得四边形的坐标,然后被要求将四边形的3个单元移到左边,然后将2个单元移到下面呢?它的新坐标是什么?

Examples
::实例

Example 1
::例1

Triangle $\begin{aligned} △ A B C \end{aligned}$ has coordinates $\begin{aligned} A (3, - 1), B (7, - 5) \end{aligned}$ and $\begin{aligned} C (- 2, - 2) \end{aligned}$ . Translate $\begin{aligned} △ A B C \end{aligned}$ to the left 4 units and up 5 units. Determine the coordinates of $\begin{aligned} △ A^{'} B^{'} C^{'} \end{aligned}$ .
::ABC三角有坐标A(3)-1、B(7)-5和C(2)-2。将ABC转换为左4个单元和上5个单元。确定A_B_C的坐标。

Graph $\begin{aligned} △ A B C \end{aligned}$ . To translate $\begin{aligned} △ A B C \end{aligned}$ , subtract 4 from each $\begin{aligned} x \end{aligned}$ value and add 5 to each $\begin{aligned} y \end{aligned}$ value of its coordinates.
::ABC. 要翻译 ABC, 从每x值中减去 4, 并在坐标的每个 y 值上增加 5 。

$\begin{aligned} A (3, - 1) \to (3 - 4, - 1 + 5) = A^{'} (- 1, 4) \\ B (7, - 5) \to (7 - 4, - 5 + 5) = B^{'} (3, 0) \\ C (- 2, - 2) \to (- 2 - 4, - 2 + 5) = C^{'} (- 6, 3) \end{aligned}$

::A(3,-1)-(3 - 4,-1+5)=A*(1,4)B(7,5)-(7,4,-5+5)=B[3,0)C(-2,2,-2)-(-2,4,-4,-2,+5)=C*(-6,3)

The rule would be $\begin{aligned} (x, y) \to (x - 4, y + 5) \end{aligned}$ .
::规则是(x,y)(x-4,y+5)。

Example 2
::例2

Using the translation $\begin{aligned} (x, y) \to (x + 2, y - 5) \end{aligned}$ , what is the image of $\begin{aligned} A (- 6, 3) \end{aligned}$ ?
::使用翻译 (x,y) (x+2,y- 5) , A(-6, 3) 的图像是什么 ?

$\begin{aligned} A^{'} (- 4, - 2) \end{aligned}$
::A*(-4,-2002)

Example 3
::例3

Graph square $\begin{aligned} S (1, 2), Q (4, 1), R (5, 4) \end{aligned}$ and $\begin{aligned} E (2, 5) \end{aligned}$ . Find the image after the translation $\begin{aligned} (x, y) \to (x - 2, y + 3) \end{aligned}$ . Then, graph and label the image.
::图形S(1,2,Q(4,1,1,R(5,4)和E(2,5,5)和E(2,5),在翻译(x,y)(x--2,y+3)后查找图像。

We are going to move the square to the left 2 and up 3.
::我们要把广场移到左边2和3

$\begin{aligned} (x, y) & \to (x - 2, y + 3) \\ S (1, 2) & \to S^{'} (- 1, 5) \\ Q (4, 1) & \to Q^{'} (2, 4) \\ R (5, 4) & \to R^{'} (3, 7) \\ E (2, 5) & \to E^{'} (0, 8) \end{aligned}$

:x,y)(x-2,y+3)S(1,2,9+3)S[1,5]S[1,5]Q(4,1,1,2,4)R(5,4)R[3,7)E(2,5)E(0,8)

Example 4
::例4

Find the translation rule for $\begin{aligned} △ T R I \end{aligned}$ to $\begin{aligned} △ T^{'} R^{'} I^{'} \end{aligned}$ .
::查找 TRI 到 TRI 的翻译规则。

Look at the movement from $\begin{aligned} T \end{aligned}$ to $\begin{aligned} T^{'} \end{aligned}$ . The translation rule is $\begin{aligned} (x, y) \to (x + 6, y - 4) \end{aligned}$ .
::翻译规则是 (x,y)(x+6,y-4)。

Review
::回顾

Use the translation $\begin{aligned} (x, y) \to (x + 5, y - 9) \end{aligned}$ for questions 1-7.
::问题1-7使用翻译(x,y)%(x+5,y-9)

What is the image of $\begin{aligned} A (- 1, 3) \end{aligned}$ ?
::A(-1)的图像是什么?

What is the image of $\begin{aligned} B (2, 5) \end{aligned}$ ?
::B(2,5)的图像是什么?

What is the image of $\begin{aligned} C (4, - 2) \end{aligned}$ ?
::C(4)-2的图像是什么?

What is the image of $\begin{aligned} A^{'} \end{aligned}$ ?
::AA的图像是什么?

What is the preimage of $\begin{aligned} D^{'} (12, 7) \end{aligned}$ ?
::D(12,7)的预景是什么?

What is the image of $\begin{aligned} A^{″} \end{aligned}$ ?
::什么是A的形象?

Plot $\begin{aligned} A, A^{'}, A^{″}, \end{aligned}$ and $\begin{aligned} A^{‴} \end{aligned}$ from the questions above. What do you notice?
::从上面的问题中绘制A、A、A、A和A,你注意到什么了?

The vertices of $\begin{aligned} △ A B C \end{aligned}$ are $\begin{aligned} A (- 6, - 7), B (- 3, - 10) \end{aligned}$ and $\begin{aligned} C (- 5, 2) \end{aligned}$ . Find the vertices of $\begin{aligned} △ A^{'} B^{'} C^{'} \end{aligned}$ , given the translation rules below.
::ABC的顶点为A(-6),B(-3)-10和C(-5),2。参照以下翻译规则,找出QAABC的顶点。

$\begin{aligned} (x, y) \to (x - 2, y - 7) \end{aligned}$
:x,y)(x-2,y-7)

$\begin{aligned} (x, y) \to (x + 11, y + 4) \end{aligned}$
:x,y)(x+11,y+4)

$\begin{aligned} (x, y) \to (x, y - 3) \end{aligned}$
:x,y)(x,y-3)

$\begin{aligned} (x, y) \to (x - 5, y + 8) \end{aligned}$
:x,y)(x-5,y+8)

$\begin{aligned} (x, y) \to (x + 1, y) \end{aligned}$
:x,y)(x+1,y)

$\begin{aligned} (x, y) \to (x + 3, y + 10) \end{aligned}$
:x,y)(x+3,y+10)

In questions 14-17, $\begin{aligned} △ A^{'} B^{'} C^{'} \end{aligned}$ is the image of $\begin{aligned} △ A B C \end{aligned}$ . Write the translation rule.
::在问题14-17中,A_B_C_是ABC的图像。写翻译规则。

Use the triangles from #17 to answer questions 18-20.
::使用 #17 的三角形回答问题 18 -20。

Find the lengths of all the sides of $\begin{aligned} △ A B C \end{aligned}$ .
::找出ABC所有侧面的长度。

Find the lengths of all the sides of $\begin{aligned} △ A^{'} B^{'} C^{'} \end{aligned}$ .
::找出AABC的两侧长度。

What can you say about $\begin{aligned} △ A B C \end{aligned}$ and $\begin{aligned} △ A^{'} B^{'} C^{'} \end{aligned}$ ? Can you say this for any translation?
::你能对"ABC"和"ABC"说些什么吗?

If $\begin{aligned} △ A^{'} B^{'} C^{'} \end{aligned}$ was the preimage and $\begin{aligned} △ A B C \end{aligned}$ was the image, write the translation rule for #14.
::如果"A"B"C"是预视图像,"ABC"是图像, 请写14号的翻译规则。

If $\begin{aligned} △ A^{'} B^{'} C^{'} \end{aligned}$ was the preimage and $\begin{aligned} △ A B C \end{aligned}$ was the image, write the translation rule for #15.
::如果"A"B"C"是预视图像,"ABC"是图像, 请写第15页的翻译规则。

Find the translation rule that would move $\begin{aligned} A \end{aligned}$ to $\begin{aligned} A^{'} (0, 0) \end{aligned}$ , for #16.
::找到将 A 移到 A {0,0 的翻译规则, 用于 # 16 。

The coordinates of $\begin{aligned} △ D E F \end{aligned}$ are $\begin{aligned} D (4, - 2), E (7, - 4) \end{aligned}$ and $\begin{aligned} F (5, 3) \end{aligned}$ . Translate $\begin{aligned} △ D E F \end{aligned}$ to the right 5 units and up 11 units. Write the translation rule.
::DEF的坐标是D(4)-2,E(7)-4和F(5,5,3),将DEF转换到右边的5个单位和11个单位。写翻译规则。

The coordinates of quadrilateral $\begin{aligned} Q U A D \end{aligned}$ are $\begin{aligned} Q (- 6, 1), U (- 3, 7), A (4, - 2) \end{aligned}$ and $\begin{aligned} D (1, - 8) \end{aligned}$ . Translate $\begin{aligned} Q U A D \end{aligned}$ to the left 3 units and down 7 units. Write the translation rule.
::四边QUAD的坐标是Q(-6)1,U(-3,7),A(4)-2和D(1,-8),将QUAD向左3个单位和向下7个单位翻译。写翻译规则。

Review (Answers)
::回顾(答复)

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。

章节大纲

常规

几何翻译

Translations ::笔译翻译

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Translations
::笔译翻译

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)