10.1平行四边形的面积

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  平行四边形的面积

  

  [Figure 1]

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  Timmy works after school mowing lawns in the neighborhood. He is preparing a work order for a job tomorrow afternoon, and needs to know the approximate size of the yard he will be mowing. The yard is a  rectangle  with equal size  right triangles  cut from both ends for a children’s play  area  and a flower garden. So, the  space  that needs mowing is a   with a base of 60 feet and a  height  of 40 feet. How can Timmy use this information to determine the total area he will be mowing?
  ::蒂米在附近的学校修剪草坪后工作。 他正在为明天下午的工作准备一份工作订单,并需要了解他将要修剪的院子的大致大小。 院子是一个矩形,为儿童的游戏区和花园两端截取相同大小的右三角形。 因此,需要修剪的空间是60英尺和40英尺高的基脚。 提米如何利用这一信息来确定他将要修剪的总面积?

  

  [Figure 2]

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  In this concept, you will learn to find the area of parallelograms given a base and a height.
  ::在此概念中, 您将学会找到一个基数和高度的平行图区域 。

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  Finding the Area of a Parallelogram
  ::寻找平行区域

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  Area  is the space that is contained within the  perimeter  of a shape. It is the surface or covering of something.
  ::区域是形状周界内的空间,是表面或某物的覆盖物。

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  The area of a  parallelogram  is related to the area of a  rectangle , a four-sided flat shape with four right triangles and  opposite  sides that are equal and  parallel . Because a rectangle is a  quadrilateral  with opposite sides that are equal and parallel, it is considered a  parallelogram .
  ::平行方形区域与矩形区域相关,即四面平方形,有四个右三角形,对面平面相同和平行。由于矩形是一个四边形,对面平面平面,对面平面平面,对面平面平面,对面平面则平面,对面平面平面平面平面平面平面平面平面。

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  Let’s begin by looking at the area of a rectangle and then using it to find the area of a parallelogram.
  ::让我们首先看看矩形区域, 然后用它找到平行图的区域。

  $$\begin{array}{r}\text{Length}\times \text{Width}=\text{Area of a Rectangle in square units}\end{array}$$

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  To find the area of a rectangle, begin by finding the measurements for its length and width.
  ::要找到矩形区域,首先要找到其长度和宽度的测量数据。

  

  [Figure 3]

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  Next, find the area by multiplying the length and the width.
  ::下一步,通过乘以长度和宽度来找到区域。

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  $\begin{array}{r}5\times 10=50\text{square inches}\end{array}$
  ::5×10 = 50 平方英寸

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  Notice that the answer is in  square  inches not just inches. If you did not know what units the rectangle was measured in, then units would be an unknown variable. Let’s call it  $\begin{array}{r}y\end{array}$ . When you multiply 5 and 10, you must also multiply the variable.
  ::请注意, 答案不单是平方英寸。 如果您不知道矩形所测量的单位是什么, 那么单位将是一个未知的变量。 我们称之为 y。 当您乘以 5 和 10 时, 您也必须乘以该变量 。

  %20%5C%5C%0A%26%26%2050y%5E2%20%5C%20%5Ctext%7Bor%7D%20%5C%2050%20%5C%20in%5E2%0A%5Cend%7Barray%7D">

  $$\begin{array}{r}\begin{array}{rcl}& & 5y\times 10y=50(y)(y)\\ & & 50y^2\text{or}50i{n}^2\end{array}\end{array}$$

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  Let’s look at a visual representation of this. The rectangle below is divided into  square units .
  ::让我们来看看这个视觉的表达方式。 下面的矩形被分为平方形。

  

  [Figure 4]

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  This rectangle has 18 units. To find the area of the rectangle, multiply the width of 2 units by the length of 9 units to get the area of 18 square units.
  ::此矩形有 18 个单位。 要找到矩形区域, 请将 2 个单位的宽度乘以 9 个单位的长度, 以获得 18 个平方单位的面积 。

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  Now, let’s look at a parallelogram.
  ::现在,让我们看看一个平行图。

  

  [Figure 5]

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  If you take off the two triangles on the ends, you can see that a parallelogram is a lot like the rectangle.
  ::如果你把两边的三角形脱下, 你可以看到平行图和矩形非常相似。

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  The big difference is in the  angles . The parallelogram does not have any right angles, and its area is found by multiplying the height and base.
  ::最大的差别在于角度。 平行图没有任何正确的角度, 其区域通过乘以高度和基数来找到 。

  $$\begin{array}{r}\text{Base}\times \text{Height}=\text{Area of a Parallelogram}\end{array}$$

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  In this parallelogram, the base is 8 units and the height is 2 units.
  ::在这一平行图中,基数为8个单位,高度为2个单位。

  $\begin{array}{r}8\times 2=16\end{array}$

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  The area of this parallelogram is 16 square units.
  ::这一平行图的面积为16平方单位。

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  Let’s take a look at another parallelogram.
  ::让我们来看看另一个平行图。

  

  [Figure 6]

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  To find the area of this parallelogram, multiply the base by the height.
  ::要找到此平行图的区域, 请将基数乘以高度 。

  $$\begin{array}{r}\begin{array}{rcl}A& =& bh\\ A& =& (3)(8)\\ A& =& 24\text{ sq. inches}\end{array}\end{array}$$

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Timmy and the yard he has to mow.
  ::早些时候,你被提米的问题 和他必须修剪的院子。

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  Timmy charges by the square foot for mowing lawns in the neighborhood. He went by his neighbor’s house earlier and got the exact measurements of the yard he will be mowing, so he can prepare the work order. The yard is shaped like a parallelogram with a base of 60 feet and a height of 40 feet. How can Timmy use this information to determine the approximate size of the yard for his work order?
  ::Timmy在附近修剪草坪时用平方英尺的平方英尺收费。 他早些时候去了邻居家,获得了他将要修剪的院子的精确测量数据,以便他能够准备工作秩序。 院子的形状像一个60英尺和40英尺高的平行图。 Timmy如何利用这一信息来确定院子的大致大小?

  

  [Figure 7]

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  To solve this problem, use the area formula for parallelograms.
  ::要解决这个问题,请使用区域公式表示平行图。

  $$\begin{array}{r}A=bh\end{array}$$

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  First, substitute the given values for base and height.
  ::首先,将给定值替换为基数和高度。

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  $\begin{array}{r}A=(60)(40)\end{array}$
  ::A=(60) (40)

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  Next, multiply the base and height.
  ::下一个,乘以基数和高度。

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  $\begin{array}{r}A=240\text{ sq. feet}\end{array}$
  ::A=240平方英尺

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  The answer is 240 sq. feet. Timmy can use this amount to determine how he will charge for mowing his neighbor’s yard.
  ::答案是240平方英尺。 提米可以用这个数额来决定他如何修剪邻居的院子。

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  Example 2
  ::例2

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  Find the area of the parallelogram with a base of 4.5 feet and a height of 2.5 feet.
  ::以4.5英尺和2.5英尺的高度为基点,找到平行图的区域。

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  To solve this problem, use the area formula for parallelograms.
  ::要解决这个问题,请使用区域公式表示平行图。

  $$\begin{array}{r}A=bh\end{array}$$

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  First, substitute the given values for base and height.
  ::首先,将给定值替换为基数和高度。

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  $\begin{array}{r}A=(4.5)(2.5)\end{array}$
  ::A = (4.5) (2.5) = (4.5) (2.5)

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  Next, multiply the base and height.
  ::下一个,乘以基数和高度。

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  $\begin{array}{r}A=11.25\text{ sq. feet}\end{array}$
  ::A=11.25平方英尺

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  The answer is 11.25 sq. feet.
  ::答案是11.25平方英尺

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  Example 3
  ::例3

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  Find the area of the parallelogram.
  ::查找平行图的区域 。

  

  [Figure 8]

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  First, substitute the given values for base and height.
  ::首先,将给定值替换为基数和高度。

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  $\begin{array}{r}A=(2)(6)\end{array}$
  ::A = (2) (6) = (2) (6)

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  Next, multiply the base and height.
  ::下一个,乘以基数和高度。

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  $\begin{array}{r}A=12sq.cm.\end{array}$
  ::A = 12 s q. c m.

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  The answer is 12 sq. cm.
  ::答案是12平方厘米

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  Example 4
  ::例4

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  Find the area of the parallelogram.
  ::查找平行图的区域 。

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  Base = 9 inches, Height = 3.5 inches
  ::基数=9英寸,高度=3.5英寸

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  First, substitute the given values for base and height.
  ::首先,将给定值替换为基数和高度。

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  $\begin{array}{r}A=(9)(3.5)\end{array}$
  ::A = (9) (3.5) = (9) (3.5)

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  Next, multiply the base and height.
  ::下一个,乘以基数和高度。

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  $\begin{array}{r}A=31.5\text{ sq. inches}\end{array}$
  ::A=31.5平方米英寸

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  The answer is 31.5 sq. inches.
  ::答案是31.5平方英寸

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  Example 5
  ::例5

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  Find the area of the parallelogram.
  ::查找平行图的区域 。

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  Base = 5 inches, Height = 3 inches
  ::基数 = 5 英寸,高度 = 3 英寸

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  First, substitute the given values for base and height.
  ::首先,将给定值替换为基数和高度。

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  $\begin{array}{r}A=(5)(3)\end{array}$
  ::A = (5) (3) = (5) (3)

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  Next, multiply the base and height.
  ::下一个,乘以基数和高度。

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  $\begin{array}{r}A=15\text{ sq. inches}\end{array}$
  ::A=15平方英寸

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  The answer is 15 sq. inches.
  
  ::答案是15平方英寸

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  Review
  ::回顾

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  Find the area of each parallelogram using the given dimensions.
  ::使用给定的维度查找每个平行图的区域 。

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  1. Base = 7 inches, Height = 4 inches
     ::基数 = 7 英寸,高度 = 4 英寸
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  2. Base = 8 cm, Height = 2 cm
     ::基数=8厘米,高度=2厘米
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  3. Base = 9 inches, Height = 4 inches
     ::基数 = 9 英寸,高度 = 4 英寸
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  4. Base = 10 feet, Height = 5 feet
     ::基地=10英尺,高度=5英尺
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  5. Base = 7 inches, Height = 6 inches
     ::基数 = 7 英寸,高度 = 6 英寸
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  6. Base = 10 meters, Height = 8 meters
     ::基数 = 10米,高度 = 8米
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  7. Base = 11 feet, Height = 9 feet
     ::基地=11英尺,高度=9英尺
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  8. Base = 12 meters, Height = 10 meters
     ::基地=12米,高度=10米
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  9. Base = 5 inches, Height = 4.5 inches
     ::基数=5英寸,高度=4.5英寸
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  10. Base = 8.5 feet, Height = 2.5 feet
      ::基数=8.5英尺,高度=2.5英尺
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  11. Base = 9.5 feet, Height = 3 feet
      ::基数=9.5英尺,高度=3英尺
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  12. Base = 6.5 feet, Height = 3.5 feet
      ::基数=6.5英尺,高度=3.5英尺
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  13. Base = 9.5 cm, Height = 2 cm
      ::基数=9.5厘米,高度=2厘米
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  14. Base = 15 feet, Height = 12 feet
      ::基地=15英尺,高度=12英尺
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  15. Base = 150 miles, Height = 20 miles
      ::基地 = 150英里 高度 = 20英里