1.12 多边形分类

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  多边形分类

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  Polygons
  ::多边形

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  A polygon is any closed, 2-dimensional figure that is made entirely of line segments that intersect at their endpoints. Polygons can have any number of sides and angles , but the sides can never be curved. The segments are called the sides of the polygons, and the points where the segments intersect are called vertices .
  ::多边形是指完全由端点交错的线段组成的任何封闭的、二维的图。多边形可以拥有任何多个侧面和角度,但两边永远无法曲线。这些区段被称为多边形的侧面,这些区段被称为顶点。

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  Polygons can be either convex or concave . The term concave refers to a cave, or the polygon is “caving in”. All stars are concave polygons.
  ::多边形可以是二次曲线,也可以是二次曲线。“二次曲线”一词是指一个洞穴,或者多边形是“上升”的。所有恒星都是二次曲线多边形。

  

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  A convex polygon does not cave in. Convex polygons look like:
  ::二次曲线多边形不向内倾斜。 二次曲线多边形看起来像 :

  

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  A diagonal is a non-side line segment that connects two vertices of a convex polygon.
  ::对角线是一个非侧线段,它连接了锥形多边形的两个顶部。

  

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  The red line segments are all diagonals . This pentagon has 5 diagonals.
  ::红色线段都是对角线。 这个五角形有 5 个对角线段 。

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  Whether a polygon is convex or concave, it is always named by the number of sides. Explore the relationship between the number of sides of a convex polygon and its diagonals. Can you complete the table?
  ::多边形是二次曲线还是二次曲线, 它总是用边数命名。 探索二次曲线多边形的两边数与其对角线之间的关系。 您能否完成表格 ?

  Polygon Name	Number of Sides	Number of Diagonals	Convex Example
  Triangle	3	0
  Quadrilateral	4	2
  Pentagon	5	5
  Hexagon	6	9
  Heptagon	7	?
  Octagon	8	?
  Nonagon	9	?
  Decagon	10	?
  Undecagon or hendecagon	11	?
  Dodecagon	12	?
  n-gon	$\begin{align*}n\end{align*}$ (where $\begin{align*}n > 12\end{align*}$ )	?

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  What if you were told how many sides a polygon has? How would you describe the polygon based on that information?
  ::如果告诉您一个多边形有多少边呢? 您如何根据这些信息描述多边形 ?

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  Examples
  ::实例

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  Example 1
  ::例1

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  Which of the figures below is a polygon?
  ::以下哪些数字是多边形?

  

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  The easiest way to identify the polygon is to identify which shapes are not polygons. $\begin{align*}B\end{align*}$ and $\begin{align*}C\end{align*}$ each have at least one curved side, so they are not be polygons. $\begin{align*}D\end{align*}$ has all straight sides, but one of the vertices is not at the endpoint , so it is not a polygon. $\begin{align*}A\end{align*}$ is the only polygon.
  ::识别多边形的最简单方法是确定哪些形状不是多边形。B和C每个形状至少有一个曲线边,因此它们不是多边形。D有所有直线边,但有一个顶点不是终点,因此它不是一个多边形。A是唯一的多边形。

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  Example 2
  ::例2

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  Determine if the shapes below are convex or concave.
  ::确定下面的形状是二次曲线还是二次曲线。

  

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  To see if a polygon is concave, look at the polygons and see if any angle “caves in” to the interior of the polygon. The first polygon does not do this, so it is convex. The other two do, so they are concave.
  ::查看多边形是否为圆锥形, 看看多边形, 看看多边形内部是否有角“ 洞穴” 。 第一个多边形不这样做, 所以它是圆锥形。 另外两个多边形是圆锥形, 所以它们是圆锥形 。

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  Example 3
  ::例3

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  Name the three polygons below by their number of sides and if it is convex or concave.
  ::下面三个多边形按其边数排列,如果是二次曲线或二次曲线的话。

  

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  The pink polygon is a concave hexagon (6 sides).
  ::粉红色多边形是六角锥形(6面)。

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  The green polygon convex pentagon (5 sides).
  ::绿色多边形锥形五角形(5面)。

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  The yellow polygon is a convex decagon (10 sides).
  ::黄色多边形是十角形( 10 边) 。

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  Example 4
  ::例4

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  Draw a 7-sided polygon, also called a heptagon.  How many diagonals does a heptagon  have?
  ::绘制一个 7 面多边形, 也称为七边形。 七面形有多少对角形?

  

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  First, draw the heptagon. Drawing in all the diagonals and counting them, we see there are 14.
  ::首先,绘制七角形。在所有对角图中绘制并计算它们,我们看到有14个。

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  Example 5
  ::例5

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  True or false: A quadrilateral is always a square .
  ::真实或虚假:四边形总是方形。

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  False. Only quadrilaterals with four congruent sides and four right angles will be squares. There are many quadrilaterals (such as , , , , etc.) that are not necessarily squares.
  ::假的。只有四边和四个正方形的四边方和四个右角是正方形。许多四边方(如 、 、 、 、 等)不一定是正方形。

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  Review
  ::回顾

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  In problems 1-6, name each polygon in as much detail as possible.
  ::在问题1-6中,请尽可能详细地列出每个多边形的名称。

  1. 
  2. 
  3. 
  4. 
  5. 
  6. 
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  7. Explain why the following figures are NOT polygons:
     ::解释为何以下数字不是多边形:
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  8. How many diagonals can you draw from one vertex of a pentagon? Draw a sketch of your answer.
     ::您可从五角形的一个顶端中绘制多少对角线? 请绘制您答案的草图 。
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  9. How many diagonals can you draw from one vertex of an octagon? Draw a sketch of your answer.
     ::您可从八边形的一个顶端中绘制多少对角线? 请绘制您答案的草图 。
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  10. How many diagonals can you draw from one vertex of a dodecagon?
      ::您可从 dodeagon 的一个顶端中绘制多少对角线 ?
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  11. Determine the number of total diagonals for an octagon, nonagon, decagon, undecagon, and dodecagon.
      ::确定八边形、九角形、十角形、十角形、十角形和十二角形的总对角数。

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  For 12-14, determine if the statement is true or false.
  ::对于12-14,确定声明是真实的还是虚假的。

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  12. A polygon must be enclosed.
      ::必须附上一个多边形。
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  13. A star is a convex polygon.
      ::恒星是锥形多边形。
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  14. A 5-point star is a decagon.
      ::五点恒星是十角星

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  Review (Answers)
  ::回顾(答复)

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  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 。

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  Resources
  ::资源