5.4 常规多边形建筑-interactive

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  常规多边形建筑

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  A regular polygon is a polygon that is equiangular and equilateral. This means that all its angles are the same measure and all its sides are the same length.
  ::一个正则多边形是一个以等角和等边形为多边形的多边形。这意味着其所有角度都是相同的度量,其两边的长度相同。

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  The most basic example of a regular polygon is an , a triangle with three congruent sides and three congruent angles. Squares are also regular polygons, because all their angles are the same $\begin{align*}(90^\circ)\end{align*}$ and all their sides are the same length. Regular polygons with five or more sides do not have special names. Instead, the word regular is used to describe them. For example, a regular hexagon is a hexagon (6 sided polygon) whose angles are all the same measure and sides are all the same length.
  ::普通多边形的最基本例子是 , 一个三角形, 有三个相容的边和三个相近的角。 方形也是普通多边形, 因为它们的角相同( 90) , 其两边的长度相同 。 具有五个或五个以上边的正多边形没有特殊名称 。 相反, 通常的词用来描述它们 。 例如, 普通的六边形是一个六边形( 6边多边形) , 其角都是相同的度量和边的长度相同 。

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  All regular polygons have rotation symmetry. This means that a rotation of less than $\begin{align*}360^\circ\end{align*}$ will carry the regular polygon onto itself. In fact, a regular $\begin{align*}n\end{align*}$ -sided polygon has rotation symmetry for any multiple of $\begin{align*}\frac{360^\circ}{n}\end{align*}$ .
  ::所有正态多边形都有旋转对称。 这意味着一个低于 360 的旋转会将正态多边形随身携带。 事实上, 一个正态的正向多边形对应于360 的任意倍数。

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  Any regular polygon can be constructed with geometry software or the appropriate tools:
  ::任何常规多边形都可以使用几何软件或适当工具建造:

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  • Compass: A device that allows you to create a circle with a given radius . Not only can compasses help you to create circles, but also they can help you to copy distances.
    ::指南针 : 一种设备, 允许您用给定半径创建圆圈 。 不仅指南针可以帮助您创建圆圈, 还可以帮助您复制距离 。
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  • Straightedge: Anything that allows you to produce a straight line . A straightedge should not be able to measure distances. An index card works well as a straightedge. You can also use a ruler as a straightedge, as long as you only use it to draw straight lines and not to measure.
    ::直线: 任何允许您生成直线的东西。 直线不应能够测量距离。 索引卡和直线一起工作。 您也可以使用标尺作为直线, 只要您只用它来绘制直线而不测量直线。
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  • Paper: When a geometric figure is on a piece of paper, the paper itself can be folded in order to construct new lines.
    ::纸张:当几何图出现在一张纸上时,纸本身可以折叠,以建造新的线条。

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  You can construct some regular polygons by hand if you remember the definitions and properties of these regular polygons.
  ::如果您记得这些普通多边形的定义和属性, 您可以手工构建一些普通多边形 。

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  Construct an Equilateral Triangle
  ::构造等边三角形

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  Use a compass and straightedge to construct an equilateral triangle.
  ::使用指南针和直角来构建一个等边三角形。

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  Consider $\begin{align*}\overline{AB}\end{align*}$ as one side of what will become equilateral triangle $\begin{align*}\Delta ABC\end{align*}$ . You need to put point $\begin{align*}C\end{align*}$ in the correct place in order to make the equilateral triangle. The distance of point $\begin{align*}C\end{align*}$ from point $\begin{align*}A\end{align*}$ or point $\begin{align*}B\end{align*}$ needs to be the same as the distance between point $\begin{align*}A\end{align*}$ and point $\begin{align*}B\end{align*}$ .
  ::将 AB 视为 Q ABC 等边三角形的一面 。 您需要将 C 点放在正确的位置, 才能建立 等边三角形 。 C 点从 A 点或 B 点的距离必须与 A 点 和 B 点 之间的距离相同 。

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  Draw a line segment $\begin{align*}AB\end{align*}$ , using a straightedge. Use a compass to measure the length of $\begin{align*}\overline{AB}\end{align*}$ .
  ::绘制直线段AB。 使用指南针测量 AB 的长度 。

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  Keeping your compass width the same, away from the vertex $\begin{align*}A\end{align*}$ , draw a little arc , which is the same length as $\begin{align*}AB\end{align*}$ . Away from the vertex $\begin{align*}B\end{align*}$ , draw another arc , which is the same length as $\begin{align*}AB\end{align*}$ .
  ::将罗盘宽度保持相同, 远离顶端 A, 绘制一个小弧, 与 AB 的长度相同。 远离顶端 B, 绘制另一个弧, 与 AB 的长度相同 。

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  The point of intersection of these two arcs is point $\begin{align*}C\end{align*}$ . Join $\begin{align*}A\end{align*}$ to $\begin{align*}C\end{align*}$ and $\begin{align*}B\end{align*}$ to $\begin{align*}C\end{align*}$ . The triangle $\begin{align*}ABC\end{align*}$ so obtained is the required triangle.
  ::这两个弧的交叉点是点C. join A至C和点B至C。 以这种方式获得的三角ABC是所需的三角。

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  Examples
  ::实例实例实例实例

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  Example 1
  ::例1

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  Use your compass to construct a circle like the one shown below on a piece of paper. Describe how to fold the paper two times in order to help you construct a square .
  ::使用您的指南针构造一个圆形, 如下面在一张纸上显示的圆形。 描述如何将纸张折叠两次, 以便帮助您构建一个正方形 。

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  Fold the circle so that the two halves overlap to create a crease that is the diameter .
  ::折叠圆圈,使两半部分重叠,形成直径的折痕。

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  Fold the circle in half again to create the perpendicular bisector of the diameter. To do this, fold so that the two endpoints of the diameter meet. The second crease will also be a diameter.
  ::将圆再折成两半, 以创建直径的直径的直径直径的直角两侧。 要做到这一点, 折成直径的两个端点相交。 第二个折叠也将是一个直径 。

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  Note that the two diameters are perpendicular to one another. Connect the four points of intersection on the circle to construct a square .
  ::请注意,两个直径是垂直的。连接圆上四个交叉点以构建一个正方形。

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  Example 2
  ::例2

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  Prove that the quadrilateral formed by the intersections of a circle with perpendicular diameters is a square.
  ::证明圆圆的交叉点形成的四边形与直径直径为直径的圆形交叉点构成的四边形是一个方形。

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    $\begin{align*}\overline{AO} \cong \overline{BO} \cong \overline{CO} \cong \overline{DO}\end{align*}$ because they are all radii of the same circle. Since $\begin{align*}\angle BOC\end{align*}$ is a right angle, $\begin{align*}\angle BOA\end{align*}$ , $\begin{align*}\angle AOD\end{align*}$ and $\begin{align*}\angle COD\end{align*}$ must also be right angles. Therefore, $\begin{align*}\angle BOC\end{align*}$ $\begin{align*}\cong\end{align*}$ $\begin{align*}\angle BOA\end{align*}$ $\begin{align*}\cong\end{align*}$ $\begin{align*}\angle AOD\end{align*}$ $\begin{align*}\cong\end{align*}$ $\begin{align*}\angle COD\end{align*}$ . This means that $\begin{align*}\Delta AOB \cong \Delta BOC \cong \Delta COD \cong \Delta DOA\end{align*}$ by $\begin{align*}SAS \ \cong\end{align*}$ . $\begin{align*}\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{DA}\end{align*}$ because they are corresponding parts of congruent triangles.
  ::由于“BOC”是一个右角度,所以“BOA”、“AOD”和“COD”必须是右角度。因此,“BOC”和“COD”是指SAS的“AOB”和“COD”之间的对应三角。

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  All four triangles are isosceles because they each have two congruent sides . This means that their base angles are congruent. Because the vertex angle of each triangle is $\begin{align*}90^\circ\end{align*}$ , the base angles of each triangle must be $\begin{align*}45^\circ ( 90 + 45 + 45 = 180 )\end{align*}$ . The four angles that make up the quadrilateral are each made of two $\begin{align*}45^\circ\end{align*}$ angles, and are each therefore $\begin{align*}90^\circ\end{align*}$ .
  ::所有四个三角都是等分形, 因为每个三角都有两个相近的边。 这意味着它们的底角是相同的。 因为每个三角的顶角是 90 , 每个三角的底角必须是 45 ( 90+ 45+45+45=180) 。 构成四边形的四个角度由两个 45 角度组成, 因此每个角度都是 90 。

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  Because the quadrilateral has four congruent sides and four $\begin{align*}90^\circ\end{align*}$ angles, it is a square.
  ::因为四边有四个相近的两面 四个90角,就是一个方形

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  Example 3
  ::例3

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  The regular hexagon below has been divided into six . What type of triangles are they? Explain.
  ::下面的普通六边形被分为六。它们是哪种三角形? 请解释 。

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  They must be equilateral triangles.
  ::它们必须是等边三角形。

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  • A full circle is $\begin{align*}360^\circ\end{align*}$ , so each angle at the center of the hexagon must be $\begin{align*}\frac{360^\circ}{6}=60^\circ\end{align*}$ . * This is also why regular hexagons demonstrate rotation symmetry at multiples of $\begin{align*}60^\circ\end{align*}$ .*
    ::全圆为 360 , 所以在六边形中心的每一角度都必须是 360 6= 60 。 * 这也是普通六边形在60 的倍数上显示旋转对称的原因。 *
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  • The six triangles are congruent , so the six segments connecting the center of the hexagon to the vertices must be congruent. This means the six triangles are all isosceles.
    ::6个三角形是相似的, 所以连接六边形中心到脊椎的6个区段必须是相似的。 这意味着 6个三角形都是等分形 。
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  • The base angles of each of the isosceles triangles must be $\begin{align*}\frac{180-60}{2}=60^\circ\end{align*}$ .
    ::每个等分三角形的基角度必须是 180-602=60 。
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  • The measure of each angle of all of the triangles is $\begin{align*}60^\circ\end{align*}$ , so all triangles are equilateral .
    ::所有三角形角度的每个角度的度量为 60 , 所以所有三角形都是等边的 。

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  Example 4
  ::例4

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  Six points have been evenly spaced around the circle below. Explain why a regular hexagon is created when these points are connected.
  ::六个点在以下圆圈上均匀的间距。 请解释为什么在这些点连接时会创建普通的六边形 。

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  Because the six points are evenly spaced , each of the segments connecting the six points must be the same length. Therefore, the polygon must be regular. Because there are six sides, it must be a regular hexagon.
  ::因为六点间距均匀, 连接六点的每个区段的长度必须相同。 因此, 多边形必须是正常的。 因为有六边, 它必须是普通的六边 。

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  Example 5
  ::例5

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  Construct a regular hexagon inscribed in a circle.
  ::构造在圆形中嵌入的正则六边形。

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  “Inscribed in a circle” means all six vertices of the hexagon are on the same circle. Start by constructing a circle and a point on the circle.
  ::“在圆内标注”是指六边形的所有六个顶点都在同一圆上。从在圆上建一个圆开始,然后在圆上建一个圆和一个点。

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  The goal is to place six points around the circle that are the same distance apart from one another as the radius of the circle. Keep your compass open to the same width as the radius of the circle and make one new mark on the circle.
  ::目标是在圆圆周围放置与圆半径相距相同的六点。使指南针保持与圆半径相同的宽度,并在圆圆上做一个新的标记。

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  Continue to make new marks around the circle that are the same distance apart from one another.
  ::继续环绕圆圈做新标记,彼此相距相同。

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  Connect the intersection points to form the regular hexagon.
  ::连接交叉点形成正则六边形。

  Summary

  播放段落 A regular polygon is a polygon that is equiangular and equilateral. This means that all its angles are the same measure and all its sides are the same length. ::一个正则多边形是一个以等角和等边形为多边形的多边形。这意味着其所有角度都是相同的度量,其两边的长度相同。 播放段落 All regular polygons have rotation symmetry. This means that a rotation of less than  will carry the regular polygon onto itself.  ::所有正则多边形都具有旋转对称性。 这意味着旋转小于将正则多边形随身携带的旋转。

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  Reviews
  ::审查

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  1. Construct an equilateral triangle.
  ::1. 构建一个等边三角形。

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  2. Construct another equilateral triangle.
  ::2. 另建一个等边三角形。

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  3. Explain why your process for constructing equilateral triangles works.
  ::3. 解释为何建造等边三角形的过程起作用。

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  4. Construct a square inscribed in a circle by making two folds.
  ::4. 以两个折叠方式在圆形中构造一个已标定的正方形。

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  5. Justify why the polygon you've created is actually a square.
  ::5. 说明为什么你所创造的多边形实际上是一个方形。

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  Use your straightedge to construct $\begin{align*}\overline{AB}\end{align*}$ .
  ::利用你的直线 建造AB'。

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  6. Construct the perpendicular bisector of $\begin{align*}\overline{AB}\end{align*}$ .
  ::6. 建构AB的垂直两侧区块。

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  7. Construct a circle with diameter $\begin{align*}\overline{AB}\end{align*}$ .
  ::7. 构造直径为AB的圆形。

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  8. Construct a square inscribed in the circle by connecting the four endpoints of the diameters.
  ::8. 通过连接直径的四个端点,构造在圆形中标定的方形。

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  9. Extend your construction to a regular octagon by bisecting each of the right angles at the center of the circle.
  ::9. 通过在圆心中心对每个右角进行双切,将构造扩展至正八角。

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  10. Construct a regular hexagon inscribed in a circle.
  ::10. 构造在圆内刻入的正六边形。

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  11. Explain why the method for constructing a regular hexagon relies on a circle.
  ::11. 解释为何构建普通六边形的方法依赖于圆。

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  12. Explain how you could extend your construction of the regular hexagon to a construction of a regular 12-gon.
  ::12. 请解释您如何将普通六边形的构造扩展至普通12边形的构造。

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  13. Construct an equilateral triangle. Explain how you could construct the circle that passes through the three points of the equilateral triangle.
  ::13. 构建一个等边三角形,解释如何构建穿过等边三角形三点的圆圈。

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  14. Given an equilateral triangle inscribed in a circle, how could you extend the construction to construct a regular hexagon?
  ::14. 鉴于以圆形标定的等边三角形,如何扩大建筑以建造正常的六边形?

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  15. Given a circle and a protractor, explain how you could create a regular pentagon.
  ::15. 如果有一个圆形和一个减速器,请解释如何创建普通的五边形。

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  16. The Pentagon building in Arlington, Virginia, is in the shape of a regular pentagon. It can be inscribed in a circle with a radius of 781.6 feet. Draw a scale diagram from the information given. How do you know that this is a scaled version of the Pentagon in Arlington? Explain.
  ::16. 弗吉尼亚州阿林顿五角大楼的形状是普通的五角大楼,可以标为圆形,半径781.6英尺,从所提供的信息中绘制一个比例图,你怎么知道这是阿灵顿五角大楼的缩放版?解释。

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  17. You have a circular piece of tin to which you are going to cut out a new stop sign for your street. Describe how you would do this.
  ::17. 你有一个圆形的锡块,你会在街道上切开一个新的停车牌,描述你将如何这样做。

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  Review (Answers)
  ::审查(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 。