Course: 7.11 关系 | The Way To Learn

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关系

Dilation
::关系

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 but the new resulting figure has the same shape as the original.
::如果两个数字是相同的形状,但不一定是相同的大小,则两个数字是相似的。创建的方法之一是用放大法。放大法使数字大或小,但新得出的数字与原始数字具有相同的形状。

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 $\begin{aligned} k \end{aligned}$ . 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 $\begin{aligned} 0 < k < 1 \end{aligned}$ .
::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 $\begin{aligned} k > 1 \end{aligned}$ .
::如果放大图像大于原始图像,则 k>1。

A dilation, or image, is always followed by a $\begin{aligned} ^{'} \end{aligned}$ .
::雕刻或图像之后总是有。

*Label It*	*Say It*
$\begin{aligned} ^{'} \end{aligned}$	“prime” (copy of the original)
$\begin{aligned} A^{'} \end{aligned}$	“a prime” (copy of point $\begin{aligned} A \end{aligned}$ )
$\begin{aligned} A^{″} \end{aligned}$	“a double prime” (second copy)

What if you enlarged or reduced a triangle without changing its shape? How could you find the scale factor by which the triangle was stretched or shrunk?
::如果您在不改变三角形形状的情况下扩大或缩小三角形呢? 您如何找到三角形被拉伸或缩缩的缩放因子 ?

Examples
::实例

Example 1
::例1

Find the perimeters of $\begin{aligned} K L M N \end{aligned}$ and $\begin{aligned} K^{'} L^{'} M^{'} N^{'} \end{aligned}$ . Compare this ratio to the scale factor.
::查找 KLMN 和 KL*M*N 的周界。比较此比例与比例系数。

The perimeter of $\begin{aligned} K L M N = 12 + 8 + 12 + 8 = 40 \end{aligned}$ . The perimeter of $\begin{aligned} K^{'} L^{'} M^{'} N^{'} = 24 + 16 + 24 + 16 = 80 \end{aligned}$ . The ratio is 80:40, which reduces to 2:1, which is the same as the scale factor.
::KLMN=12+8+12+8+8=40的周界。KLQN+24+16+24+16+16=80的周界。该比率为80:40,降至2:1,与比例系数相同。

Example 2
::例2

$\begin{aligned} △ A B C \end{aligned}$ is a dilation of $\begin{aligned} △ D E F \end{aligned}$ . If $\begin{aligned} P \end{aligned}$ is the center of dilation, what is the scale factor?
::ABC 是 QDEF 的扩展。如果 P 是放大中心, 缩放系数是什么 ?

Because $\begin{aligned} △ A B C \end{aligned}$ is a dilation of $\begin{aligned} △ D E F \end{aligned}$ , then $\begin{aligned} △ A B C \sim △ D E F \end{aligned}$ . The scale factor is the ratio of the sides. Since $\begin{aligned} △ A B C \end{aligned}$ is smaller than the original, $\begin{aligned} △ D E F \end{aligned}$ , the scale factor is going to be less than one, $\begin{aligned} \frac{12}{20} = \frac{3}{5} \end{aligned}$ .
::因为 ABC 是 QDEF 的乘数, 然后是 ABC 的乘数。比例系数是两边之比。由于 ABC 小于原来的 QDEF, 比例系数将小于 1, 1220=35 。

If $\begin{aligned} △ D E F \end{aligned}$ was the dilated image, the scale factor would have been $\begin{aligned} \frac{5}{3} \end{aligned}$ .
::如果“DEF”是放大图像,则比例系数为53。

Example 3
::例3

The center of dilation is $\begin{aligned} P \end{aligned}$ and the scale factor is 3.
::膨胀的中心是P 比例系数是3

Find $\begin{aligned} Q^{'} \end{aligned}$ .
::找寻... ... 。

If the scale factor is 3 and $\begin{aligned} Q \end{aligned}$ is 6 units away from $\begin{aligned} P \end{aligned}$ , then $\begin{aligned} Q^{'} \end{aligned}$ is going to be $\begin{aligned} 6 \times 3 = 18 \end{aligned}$ units away from $\begin{aligned} P \end{aligned}$ . The dilated image will be on the same line as the original image and center.
::如果比例系数为 3 和 Q 是 6 个单位离 P 以外, 则将 6x3= 18 个单位离 P 以外。放大的图像将与原始图像和中心线相同。

Example 4
::例4

Using the picture above, change the scale factor to $\begin{aligned} \frac{1}{3} \end{aligned}$ .
::使用上述图象,将比额表因数改为13。

Find $\begin{aligned} Q^{″} \end{aligned}$ using this new scale factor.
::使用此新的比额表因数查找 {} 。

The scale factor is $\begin{aligned} \frac{1}{3} \end{aligned}$ , so $\begin{aligned} Q^{″} \end{aligned}$ is going to be $\begin{aligned} 6 \times \frac{1}{3} = 2 \end{aligned}$ units away from $\begin{aligned} P \end{aligned}$ . $\begin{aligned} Q^{″} \end{aligned}$ will also be collinear with $\begin{aligned} Q \end{aligned}$ and center.
::缩放系数为 13, 因此将会是 6x13=2 单位从 P. 。也将是 Q 和 Central 的圆线。

Example 5
::例5

$\begin{aligned} K L M N \end{aligned}$ is a rectangle . If the center of dilation is $\begin{aligned} K \end{aligned}$ and $\begin{aligned} k = 2 \end{aligned}$ , draw $\begin{aligned} K^{'} L^{'} M^{'} N^{'} \end{aligned}$ .
::KLMN 是一个矩形。如果放大中心是 K 和 k=2, 则绘制 KL\ MN 。

If $\begin{aligned} K \end{aligned}$ is the center of dilation, then $\begin{aligned} K \end{aligned}$ and $\begin{aligned} K^{'} \end{aligned}$ will be the same point. From there, $\begin{aligned} L^{'} \end{aligned}$ will be $\begin{aligned} 8 \end{aligned}$ units above $\begin{aligned} L \end{aligned}$ and $\begin{aligned} N^{'} \end{aligned}$ will be 12 units to the right of $\begin{aligned} N \end{aligned}$ .
::如果 K 是放大中心, K 和 K 将是同一点。从那里, L 上方的 L 和 N 上方的 8 个单位将 12 个单位到 N 右方。

Review
::回顾

For the given shapes, draw the dilation, given the scale factor and center.
::对于给定形状, 绘制比照, 并给定比例系数和中心。

$\begin{aligned} k = 3.5 \end{aligned}$ , center is $\begin{aligned} A \end{aligned}$
::k=3.5, 中心为 A

$\begin{aligned} k = 2 \end{aligned}$ , center is $\begin{aligned} D \end{aligned}$
::k=2, 中心为 D

$\begin{aligned} k = \frac{3}{4} \end{aligned}$ , center is $\begin{aligned} A \end{aligned}$
::k=34, 中心为 A

$\begin{aligned} k = \frac{2}{5} \end{aligned}$ , center is $\begin{aligned} A \end{aligned}$
::k=25, 中心为 A

In the four questions below, you are told the scale factor. Determine the dimensions of the dilation. In each diagram, the black figure is the original and $\begin{aligned} P \end{aligned}$ is the center of dilation.
::在以下四个问题中,您会被告知比例系数。确定放大的尺寸。在每个图表中,黑图是原始的,P是放大的中心。

$\begin{aligned} k = 4 \end{aligned}$
::k=4 k=4

$\begin{aligned} k = \frac{1}{3} \end{aligned}$
::k=13 (k=13)

$\begin{aligned} k = 2.5 \end{aligned}$
::k=2.5 (k=2.5)

$\begin{aligned} k = \frac{1}{4} \end{aligned}$
::k=14

In the three questions below, find the scale factor, given the corresponding sides. In each diagram, the black figure is the original and $\begin{aligned} P \end{aligned}$ is the center of dilation.
::在下面的三个问题中, 找到相应的边边的比值系数。在每一图表中, 黑色数字为原始数字, P 是放大中心。

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

关系

Dilation ::关系

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)