Course: 7.12 协调计划内的关系

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坐标平面的差错
Dilation in the Coordinate Plane
::坐标平面的差错

Two figures are similar if they are the same shape but not necessarily the same size. One way to create is by dilating. A dilation makes a figure larger or smaller such that the new image has the same shape as the original.
::如果两个数字的形状相同,但不一定是相同的大小,则两个数字是相似的。创建的方法之一是拉平。放大使一个数字大或小, 使新图像与原始图像的形状相同。

Dilation: An enlargement or reduction of a figure that preserves shape but not size. All dilations are similar to the original figure.
::膨胀: 一个保存形状而非大小的数字的扩大或缩小。所有膨胀都与原始数字相似。

Dilations have a center and a scale factor . The center is the point of reference for the dilation and the scale factor tells us how much the figure stretches or shrinks. A scale factor is labeled $\begin{align*}k\end{align*}$ . Only positive scale factors , k, will be considered in this text.
::缩放有一个中心和一个比例系数。中心是放大的参考点, 缩放系数告诉我们数字拉伸或缩放的大小。标有缩放系数 k 的缩放系数。只有正缩放系数 k 才会在本文本中加以考虑。

If the dilated image is smaller than the original, then $\begin{align*}0 < k < 1\end{align*}$ .
::如果放大图像小于原始图像,则 0<k<1>。

If the dilated image is larger than the original, then $\begin{align*}k>1\end{align*}$ .
::如果放大图像大于原始图像,则 k>1。

To dilate something in the coordinate plane , multiply each coordinate by the scale factor. This is called mapping . For any dilation the mapping will be $\begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}$ . In this text, the center of dilation will always be the origin .
::要放大坐标平面上的东西, 请将每个坐标乘以比例系数。这称为映射。对于任何映射来说, 映射将会是 (x, y) {( kx, ky) 。在此文本中, 映射的中心总是源头。

What if you were given the coordinates of a figure and were asked to dilate that figure by a scale factor of 2? How could you find the coordinates of the dilated figure?
::如果给您一个数字的坐标, 并被要求将该数值扩大为 2 的比值呢? 您如何找到这个比值的坐标 ?

Examples
::实例

For Examples 1 and 2, use the following instructions:
::关于例1和例2,请使用以下说明:

Given $\begin{align*}A\end{align*}$ and the scale factor, determine the coordinates of the dilated point, $\begin{align*}A'\end{align*}$ . You may assume the center of dilation is the origin. Remember that the mapping will be $\begin{align*}(x, y) \rightarrow (kx, ky)\end{align*}$ .
::给定 A 和缩放系数 , 请确定膨胀点的坐标 A 。您可以假设放大中心为源。请记住映射是 (x,y) 和 (kx, ky) 。

Example 1
::例1

$\begin{align*}A(-4, 6), k = 2\end{align*}$
::A(-4,6)k=2

$\begin{align*}A'(-8, 12)\end{align*}$
::A_(-8,12)

Example 2
::例2

$\begin{align*}A(9, -13), k = \frac{1}{2}\end{align*}$
::A(9)-13,k=12

$\begin{align*}A'(4.5, -6.5)\end{align*}$
::A/(4.5,-6.5)

Example 3
::例3

Quadrilateral $\begin{align*}EFGH\end{align*}$ has vertices $\begin{align*}E(-4, -2), F(1, 4), G(6, 2)\end{align*}$ and $\begin{align*}H(0, -4)\end{align*}$ . Draw the dilation with a scale factor of 1.5.
::EFGH四边形的E(-4),-2,F(1,4,G(6,2)和H(0)-4)的顶点是E(-4),F(1,4,G(6,2)和H(0)-4。

Remember that to dilate something in the coordinate plane, multiply each coordinate by the scale factor.
::记住要放大坐标平面上的东西, 将每个坐标乘以比例系数。

For this dilation, the mapping will be $\begin{align*}(x, y) \rightarrow (1.5x, 1.5y)\end{align*}$ .
::对于此放大, 映射将是 (x,y) ( 1. 5x, 1. 5y) 。

$\begin{aligned} E (- 4, - 2) \to (1.5 (- 4), 1.5 (- 2)) \to E^{'} (- 6, - 3) \\ F (1, 4) \to (1.5 (1), 1.5 (4)) \to F^{'} (1.5, 6) \\ G (6, 2) \to (1.5 (6), 1.5 (2)) \to G^{'} (9, 3) \\ H (0, - 4) \to (1.5 (0), 1.5 (- 4)) \to H^{'} (0, - 6) \end{aligned}$
$\begin{align*}& E(-4, -2) \rightarrow (1.5(-4), 1.5(-2)) \rightarrow E'(-6, -3)\\ & F(1, 4) \rightarrow (1.5(1), 1.5(4)) \rightarrow F'(1.5, 6)\\ & G(6, 2) \rightarrow (1.5(6), 1.5(2)) \rightarrow G'(9,3)\\ & H(0, -4) \rightarrow (1.5(0),1.5(-4)) \rightarrow H'(0, -6)\end{align*}$
::E(4)-(4)-(4)-(4)-(4)-(5)-(4)-(5)-(5)-(4)-(6)-(6)-(3)-(1,4)-(1,4)-(1,4,(1),1.5(4))-(F)-(5,6)-(6)-(6)-(6)-(5)-(6)-(6)-(6)-(6)-(6)-(6)-(6)-(6)-(6)-(6)-(6)-(9,(6)-(9)-(9)-(9)-(9)-(0)-(5)-(5)-(5)-(4)-(0)-(0)-(6)-(0)-(6)-(6)

In the graph above, the blue quadrilateral is the original and the red image is the dilation.
::在上图中,蓝色四边形为原形,红色图像为放大图。

Example 4
::例4

Determine the coordinates of $\begin{align*}\triangle ABC\end{align*}$ and $\begin{align*}\triangle A' B' C'\end{align*}$ and find the scale factor.
::确定 ABC 和 ABC 的坐标,并查找比额表系数。

The coordinates of the vertices of $\begin{align*}\triangle ABC\end{align*}$ are $\begin{align*}A(2, 1)\end{align*}$ , $\begin{align*}B(5, 1)\end{align*}$ and $\begin{align*}C(3, 6)\end{align*}$ . The coordinates of the vertices of $\begin{align*}\triangle A'B'C'\end{align*}$ are $\begin{align*}A'(6, 3), B'(15, 3)\end{align*}$ and $\begin{align*}C'(9, 18)\end{align*}$ . Each of the corresponding coordinates are three times the original, so $\begin{align*}k = 3\end{align*}$ .
::ABC的顶部坐标为A(2)、1、B(5)、1和C(3、6),“A_B_C”的顶部坐标为A[6、3、B{15、3和C[9、18]。

Example 5
::例5

Show that dilations preserve shape by using the distance formula. Find the lengths of the sides of both triangles in Example B.
::使用距离公式显示膨胀保存形状。在例B中查找两个三角形边的长度。

$\begin{aligned} \underline{△ A B C} & \underline{△ A^{'} B^{'} C^{'}} \\ A B = \sqrt{(2 - 5)^{2} + (1 - 1)^{2}} = \sqrt{9} = 3 & A^{'} B^{'} = \sqrt{(6 - 15)^{2} + (3 - 3)^{2}} = \sqrt{81} = 9 \\ A C = \sqrt{(2 - 3)^{2} + (1 - 6)^{2}} = \sqrt{26} & A^{'} C^{'} = \sqrt{(6 - 9)^{2} + (3 - 18)^{2}} = 3 \sqrt{26} \\ C B = \sqrt{(3 - 5)^{2} + (6 - 1)^{2}} = \sqrt{29} & C^{'} B^{'} = \sqrt{(9 - 15)^{2} + (18 - 3)^{2}} = 3 \sqrt{29} \end{aligned}$
$\begin{align*}& \underline{\triangle ABC} && \underline{\triangle A'B'C'}\\ & AB = \sqrt{(2-5)^2 + (1-1)^2} = \sqrt{9} =3 && A'B' = \sqrt{(6-15)^2 + (3-3)^2} = \sqrt{81} = 9\\ & AC = \sqrt{(2-3)^2 + (1-6)^2} = \sqrt{26} && A'C' = \sqrt{(6-9)^2+(3-18)^2} = 3 \sqrt{26}\\ & CB = \sqrt{(3-5)^2 + (6-1)^2} = \sqrt{29} && C'B' = \sqrt{(9-15)^2 +(18-3)^2} = 3 \sqrt{29}\end{align*}$
::ABC=2-5,2+(1-1)2=9=3AB*B*(6-15)2+(3-3-3)2=81=9AC=(2-3)2+(1-6)2=26A_C*(6-9)2+(3-18)2=326CB=(3-5)2+(6-1)2+(6-1)2=29C_B*(9-15)2+(18-3)2=329

From this, we also see that all the sides of $\begin{align*}\triangle A'B'C'\end{align*}$ are three times larger than $\begin{align*}\triangle ABC\end{align*}$ .
::我们从中也看到,`A'B'C'两边的面积比`ABC'大三倍。

Review
::回顾

Given $\begin{align*}A\end{align*}$ and $\begin{align*}A'\end{align*}$ , find the scale factor. You may assume the center of dilation is the origin.
::鉴于 A 和 A , 找到比例系数。您可以假设放大中心是源。

$\begin{align*}A(8, 2), A'(12, 3)\end{align*}$
::A(8,2),A[12,3),A[8,2,A][12,3]

$\begin{align*}A(-5, -9), A'(-45, -81)\end{align*}$
::A(-5)-(9)-(A)-(45)-(81)

$\begin{align*}A(22, -7), A(11, -3.5)\end{align*}$
::A(22,-7),A(11,-3.5)

The origin is the center of dilation. Draw the dilation of each figure, given the scale factor.
::来源是放大中心。根据比例系数绘制每个图的放大系数。

$\begin{align*}A(2, 4), B(-3, 7), C(-1, -2); k = 3\end{align*}$
::A(2,4),B(-3,7),C(-1,-2),k=3

$\begin{align*}A(12, 8), B(-4, -16), C(0, 10); k = \frac{3}{4}\end{align*}$
::A(12,8),B(-4,-16),C(0,10);k=34

Multi-Step Problem Questions 6-9 build upon each other.
::多重问题6-9是相辅相成的。

Plot $\begin{align*}A(1, 2), B(12, 4), C(10, 10)\end{align*}$ . Connect to form a triangle.
::绘制 A(1, 2, B( 12, 4, 4, C( 10, 10) 10) 。连接成三角形。

Make the origin the center of dilation. Draw 4 rays from the origin to each point from #21. Then, plot $\begin{align*}A'(2, 4), B'(24, 8), C'(20, 20)\end{align*}$ . What is the scale factor?
::使源成为扩展的中心。从源到点绘制4个射线, 从 #21 到 4。然后, 绘图 A `( 2 , 4 ), B }( 24, 8), C `( 20, 20) 。缩放系数是什么 ?

Use $\begin{align*}k =4\end{align*}$ , to find $\begin{align*}A''B''C''\end{align*}$ . Plot these points.
::使用 k=4 来查找 ABC。绘制这些点。

What is the scale factor from $\begin{align*}A'B'C'\end{align*}$ to $\begin{align*}A''B''C''\end{align*}$ ?
::A_B_C_B_C_B_B_C_B_C_B_C_BAR_B_C_BAR_B_C_BAR_BAR_B_C_BAR_BAR_B_C_BAR_B_C_BAR_BAR_BAR_C_BAR_BAR_C_BAR_B_C_BAR_B_C_C_BAR_C_BAR_B_C_BAR_C_B_C_B_C_C_B_C_B_C_C_B_C_C_B_C_C_B_C_C_B_C_C_B_C_C_C_B_C_C_C_B_C_C_C_B_C_C_B_C_C_B_C_C_C_B_C_C_B_C_B_C_C_

If $\begin{align*}O\end{align*}$ is the origin, find the following lengths (using 6-9 above). Round all answers to the nearest hundredth.
::如果 O 是源, 请找到以下长度( 使用以上 6- 9 ) 。所有答案都回合到最近的一百个。

$\begin{align*}OA\end{align*}$
::OA 组织

$\begin{align*}AA'\end{align*}$
::AA___________________________________________________________________________________________

$\begin{align*}AA''\end{align*}$
::AA_________________________________________________________________________________________________________________________________________

$\begin{align*}OA'\end{align*}$
::OA____________________________________________________________________________

$\begin{align*}OA''\end{align*}$
::OA

$\begin{align*}AB\end{align*}$
::AB

$\begin{align*}A'B'\end{align*}$
::A_B_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{align*}A''B''\end{align*}$
::

Compare the ratios $\begin{align*}OA:OA'\end{align*}$ and $\begin{align*}AB:A'B'\end{align*}$ . What do you notice? Why do you think that is?
::比较OA:OA`和AB:AB:BA`比率。你注意到什么?你觉得这是为什么?

Compare the ratios $\begin{align*}OA:OA''\end{align*}$ and $\begin{align*}AB:A''B''\end{align*}$ . What do you notice? Why do you think that is?
::比较OA:OA和AB:AB。你注意到什么?你为什么认为那是?

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。

Section outline

General

坐标平面的差错

Dilation in the Coordinate Plane ::坐标平面的差错

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Dilation in the Coordinate Plane
::坐标平面的差错

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)