课程： 10.7 类似多边形的面积和周边

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类似多边形的面积和周边
Area and Perimeter of Similar Polygons
::类似多边形的面积和周边

Polygons are similar when their are equal and their corresponding sides are in the same proportion . Just as their corresponding sides are in the same proportion, perimeters and areas of similar polygons have a special relationship.
::多边形在等值和对应方比例相同时是相似的,正如其对应方比例相同一样,周边和类似多边形的区域有着特殊的关系。

Perimeters: The ratio of the perimeters is the same as the scale factor . In fact, the ratio of any part of two similar shapes ( diagonals , , midsegments, , etc.) is the same as the scale factor.
::Perimeter: 周边区域的比例与比例系数相同。事实上, 两种类似形状( 对角、、中间部分、等) 的任何部分的比例与比例系数相同。

Areas: If the scale factor of the sides of two similar polygons is $\begin{aligned} \frac{m}{n} \end{aligned}$ , then the ratio of the areas is $\begin{aligned} {(\frac{m}{n})}^{2} \end{aligned}$ ( Area of Similar Polygons Theorem ). You square the ratio because area is a two-dimensional measurement.
::区域 : 如果两个相似多边形的边的比重系数是毫尼,那么区域的比率是(mn)2 (类似多边形理论区域的区域)。您将比例对齐,因为区域是二维测量。

What if you were given two similar triangles and told what the scale factor of their sides was? How could you find the ratio of their perimeters and the ratio of their areas?
::假若你们获得两个相似的三角形,并被告知两边的比重,你们怎么发现两边的比重和面积的比重呢?

Examples
::实例

Example 1
::例1

The two below are similar. Find the scale factor and the ratio of the perimeters and verify that the two results are the same.
::找出周界的比重系数和比例,并核实这两个结果是否相同。

The scale factor is $\begin{aligned} \frac{16}{24} = \frac{2}{3} \end{aligned}$ .
::比额表系数为1624(2005)=23。

$\begin{aligned} P_{s m a l l} & = 2 (10) + 2 (16) = 52 u n i t s \\ P_{l a r g e} & = 2 (15) + 2 (24) = 78 u n i t s \end{aligned}$

::Pmall=2(10)+2(16)=52单位Pmall=2(15)+2(24)=78单位

The ratio of the perimeters is $\begin{aligned} \frac{52}{78} = \frac{2}{3} \end{aligned}$ .
::周边的比重是5278=23。

Example 2
::例2

Find the area of each rectangle from Example 1. Then, find the ratio of the areas and verify that it fits the Area of Similar Polygons Theorem.
::从例1中查找每个矩形的区域,然后找出区域的比例,并核实其是否与类似多边形理论区域相符。

$\begin{aligned} A_{s m a l l} & = 10 \cdot 16 = 160 u n i t s^{2} \\ A_{l a r g e} & = 15 \cdot 24 = 360 u n i t s^{2} \end{aligned}$

::小型=1016=160单位2 大型=1524=360单位2

The ratio of the areas would be $\begin{aligned} \frac{160}{360} = \frac{4}{9} \end{aligned}$ .
::这些地区的比例为160360=49。

The ratio of the sides, or scale factor is $\begin{aligned} \frac{2}{3} \end{aligned}$ and the ratio of the areas is $\begin{aligned} \frac{4}{9} \end{aligned}$ . Notice that the ratio of the areas is the square of the scale factor.
::边比或比例系数为23,区域比率为49。请注意,区域比率是比例系数的平方。

Example 3
::例3

Find the ratio of the areas of the rhombi below. The rhombi are similar.
::查找以下rhombi地区的比例。rhombi相似。

Find the ratio of the sides and square it.
::找出两边的比重,然后平方

$\begin{aligned} {(\frac{3}{5})}^{2} = \frac{9}{25} \end{aligned}$

Example 4
::例4

Two are similar. If the scale factor is $\begin{aligned} \frac{3}{4} \end{aligned}$ and the area of the smaller trapezoid is $\begin{aligned} 81 c m^{2} \end{aligned}$ , what is the area of the larger trapezoid?
::两种相似。如果比例系数是34, 较小捕捉类的面积是81厘米, 那么较大的捕捉类的面积是多少?

First, the ratio of the areas is $\begin{aligned} {(\frac{3}{4})}^{2} = \frac{9}{16} \end{aligned}$ . Now, we need the area of the larger trapezoid. To find this, set up a proportion using the area ratio.
::首先,区域比例是(342=916),现在,我们需要更大的类区域。要找到这个区域,请使用区域比例设定一个比例。

$\begin{aligned} \frac{9}{16} = \frac{81}{A} \to 9 A & = 1296 \\ A & = 144 c m^{2} \end{aligned}$

::916=81A_9A=1296A=144平方厘米2

Example 5
::例5

Two triangles are similar. The ratio of the areas is $\begin{aligned} \frac{25}{64} \end{aligned}$ . What is the scale factor?
::两个三角形相似。区域的比例是 2564。缩放系数是什么 ?

The scale factor is $\begin{aligned} \sqrt{\frac{25}{64}} = \frac{5}{8} \end{aligned}$ .
::比额表系数为2564=58。

Example 6
::例6

Using the ratios from Example 5, find the length of the base of the smaller triangle if the length of the base of the larger triangle is 24 units.
::使用例5中的比率,如果较大三角的基数为24个单位,则可以找到较小三角基数的长度。

Set up a proportion using the scale factor.
::使用比额表因数设定比例。

$\begin{aligned} \frac{5}{8} = \frac{b}{24} \to 8 b & = 120 \\ b & = 15 u n i t s \end{aligned}$

::58=b24=8b=120b=15个单位

Review
::回顾

Determine the ratio of the areas, given the ratio of the sides of a polygon.
::根据多边形两边的比例确定区域比例。

$\begin{aligned} \frac{3}{5} \end{aligned}$

$\begin{aligned} \frac{1}{4} \end{aligned}$

$\begin{aligned} \frac{7}{2} \end{aligned}$

$\begin{aligned} \frac{6}{11} \end{aligned}$

Determine the ratio of the sides of a polygon, given the ratio of the areas.
::根据区域比例确定多边形两边的比例。

$\begin{aligned} \frac{1}{36} \end{aligned}$

$\begin{aligned} \frac{4}{81} \end{aligned}$

$\begin{aligned} \frac{49}{9} \end{aligned}$

$\begin{aligned} \frac{25}{144} \end{aligned}$

This is an equilateral triangle made up of 4 congruent equilateral triangles.
::这是一个等边三角形,由4个等边三角形组成。

What is the ratio of the areas of the large triangle to one of the small triangles?
::大三角形区域与小三角形区域的比例是多少?

What is the scale factor of large to small triangle?
::从大到小三角的尺度因素是什么?

If the area of the large triangle is $\begin{aligned} 20 u n i t s^{2} \end{aligned}$ , what is the area of a small triangle?
::如果大三角的面积是20单位2,小三角的面积是多少?

If the length of the altitude of a small triangle is $\begin{aligned} 2 \sqrt{3} \end{aligned}$ , find the perimeter of the large triangle.
::如果一个小三角形的高度长度为23,请找到大三角形的周边。

Find the perimeter of the large square and the blue square.
::找到大广场和蓝色广场的周界

Find the scale factor of the blue square and large square.
::查找蓝色正方形和大正方形的比重系数。

Find the ratio of their perimeters.
::找出周边的比重

Find the area of the blue and large squares.
::找到蓝方和大方的面积

Find the ratio of their areas.
::找出他们所在地区的比例。

Find the length of the diagonals of the blue and large squares. Put them into a ratio. Which ratio is this the same as?
::查找蓝色和大方形的对角长度。将其设置成一个比例。哪个比例和这个比例相同 ?

Two rectangles are similar with a scale factor of $\begin{aligned} \frac{4}{7} \end{aligned}$ . If the area of the larger rectangle is $\begin{aligned} 294 i n^{2} \end{aligned}$ , find the area of the smaller rectangle.
::如果较大的矩形区域为294英寸,则找到较小的矩形区域。

Two triangles are similar with a scale factor of $\begin{aligned} \frac{1}{3} \end{aligned}$ . If the area of the smaller triangle is $\begin{aligned} 22 f t^{2} \end{aligned}$ , find the area of the larger triangle.
::两个三角形相似,比例系数为13。如果小三角形的面积为22平方英尺,请找到大三角形的面积。

The ratio of the areas of two similar squares is $\begin{aligned} \frac{16}{81} \end{aligned}$ . If the length of a side of the smaller square is 24 units, find the length of a side in the larger square.
::两个类似方块的区域之比为1681。如果小方块的一面长度为24个单位,则在较大的方块中找到一面的长度。

The ratio of the areas of two right triangles is $\begin{aligned} \frac{4}{9} \end{aligned}$ . If the length of the hypotenuse of the larger triangle is 48 units, find the length of the smaller triangle’s hypotenuse.
::两个右三角的面积比是49,如果较大的三角的下限长度为48个单位,则找到小三角的下限长度。

Questions 23-26 build off of each other. You may assume the problems are connected.
::问题23 -26互为补充你可以认为问题是相互联系的

Two similar rhombi have areas of $\begin{aligned} 72 u n i t s^{2} \end{aligned}$ and $\begin{aligned} 162 u n i t s^{2} \end{aligned}$ . Find the ratio of the areas.
::两个类似的Rhombi有72个单位2和162个单位2的地区。

Find the scale factor.
::找出比额表系数。

The diagonals in these rhombi are congruent. Find the length of the diagonals and the sides.
::rhombi 的对角线是相似的。找出对角线和两边的长度。

What type of rhombi are these quadrilaterals?
::这些四边形是什么样的罗姆比?

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

类似多边形的面积和周边

Area and Perimeter of Similar Polygons ::类似多边形的面积和周边

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)

Area and Perimeter of Similar Polygons
::类似多边形的面积和周边

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Review
::回顾

Review (Answers)
::回顾(答复)