5.2 线段和角的分线段和角-interactive

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  线段和角的分形

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  A construction is similar to a drawing in that it produces a visual outcome. However, while drawings are often just rough sketches that help to convey an idea, constructions are step-by-step processes used to create accurate geometric figures. To create a construction by hand, there are a few tools that you can use:
  ::构造与图纸相似,因为它能产生视觉结果。 然而,图纸往往只是有助于传递想法的粗略草图,而图纸则是用于创建准确几何数字的逐步过程。 要亲手创建一个构造,您可以使用以下几个工具:

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  1. 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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  2. Straightedge: Anything that allows you to produce a straight line .  A  straightedge is not used  to measure distances when creating constructions. 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。 创建构造时, 直线不用于测量距离。 索引卡和直线工作。 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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  3. 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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  A few important terms:
  ::几个重要术语:

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  • To bisect a segment or an angle means to divide it into two congruent parts.
    ::将一个段或一个角分成两个相容部分。
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  • A bisector of a line segment will pass through the midpoint of the line segment.
    ::一条线段的一个分区将通过线段的中点。
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  • A perpendicular bisector of a segment passes through the midpoint of the line segment and is perpendicular to the line segment.
    ::一个线段的垂直两分形穿过线段的中点,与线段垂直。

  

  INTERACTIVE

  Bisectors of Line Segments and Angles

  

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  Move the red point to explore different bisectors of the line segment  $\overline{AC}$ .
  ::移动红点以探索线段 AC 段的不同分区 。

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  In the interactive above, $\begin{array}{r}\stackrel{\leftrightarrow }{DE}\end{array}$  is the perpendicular bisector of $\begin{array}{r}\overline{AC}\end{array}$ , so $\begin{array}{r}\overline{AB}\cong \overline{BC}\end{array}$ and $\begin{array}{r}\overline{AC}\mathrm{\perp }\stackrel{\leftrightarrow }{DE}\end{array}$ when $\begin{array}{r}\mathrm{\angle }DBC={90}^{\circ }.\end{array}$
  ::在上述互动中,DE是AC的直角两侧部分,所以ABBC和ACDEDDCC=90。

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  In order to construct bisectors of segments and angles, it's helpful to remember some relevant theorems:
  ::为了建立区段和角度的分区, 记住一些相关的理论很有帮助:

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  Any point on the perpendicular bisector of a line segment will be equidistant from the endpoints of the line segment. This means that one way to find the perpendicular bisector of a segment (such as $\begin{array}{r}\overline{AB}\end{array}$ below) is to find two points that are equidistant from the endpoints of the line segment (such as $\begin{array}{r}C\end{array}$ and $\begin{array}{r}D\end{array}$ below) and connect them.
  ::直线段直角两分线的任何点与直线段端点的距离相等。 这意味着找到一个直线段端点(如AB ' below)的直角两分点的一个方法就是从直线段端点(如下文C和D)找到两个对等点并将其连接。

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  Any point on the angle bisector of an angle will be equidistant from the rays that create the angle. This means that one way to find the angle bisector of an angle (such as $\begin{array}{r}\mathrm{\angle }BAC\end{array}$ below) is to find two points that are equidistant from the rays that create the angle (such as points $\begin{array}{r}A\end{array}$ and $\begin{array}{r}D\end{array}$ below).
  ::角角角角的角角两部分上的任何点将从创建角的射线中等距。 这意味着找到角角角角两部分角的一个方法(例如以下的“BAC”)就是从创建角的射线中找到两个相距相等的点(例如下面的A点和D点)。

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  Bisecting a Line Segment by Paper Folding
  ::以纸张折叠方式对线条段进行双切

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  Draw a line segment like the one below on a piece of paper. Bisect the line by folding the paper.
  ::绘制线条段, 和下面的纸块一样。 将线条折叠成两部分 。

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  (This starting line is often referred to as a 'working line'.)
  :这一起始线通常称为“工作线 ” 。 )

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  To find the perpendicular bisector, fold the paper  and flatten it such that the endpoints lie on top of each other. By doing this, you have matched the two halves of the line segment exactly.
  ::要找到垂直的双向区块, 折叠纸并平整它, 使端点位于彼此的顶端。 通过这样做, 您已经完全匹配了线段的两半 。

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  Any point on the fold is  equidistant from both endpoints, because the fold was made when endpoints  were in the same place! This means that the crease made by the fold will be the perpendicular bisector of the line segment.
  ::折叠中的任何点在两个端点上的位置都相等, 因为折叠是当端点位于同一地点时做出的! 这意味着折叠的折叠将是直线段的直角两侧部分。

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  Construct a Perpendicular Bisector
  ::构造一个垂直双向扇形扇形

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  Draw a line segment similar to the one labeled $\begin{array}{r}\overline{AB}\end{array}$   below, then bisect it using a compass.
  ::绘制一个类似于下面标签为AB的线条段,然后用指南针将其双切。

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  Place the compass at one endpoint of the line segment. Adjust the width of the compass to be slightly longer than half the line segment length. With $\begin{array}{r}A\end{array}$ as the center , draw a circle . Similarly, with $\begin{array}{r}B\end{array}$ as the center and the same compass width, draw another circle to intersect the first. If the circles are too big to fit on the paper, draw the portion of the circle that fits on the paper.
  ::将罗盘放在线段的一个端点上。 调整罗盘宽度, 使其略大于线段长度的一半。 A作为中心, 绘制一个圆形。 同样, B作为中心, 和相同的罗盘宽度, 画另一个圆来交叉第一个圆形。 如果圆圈太大, 无法适应纸张, 请绘制适合纸张的圆圈部分 。

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  The two points $\begin{array}{r}P\end{array}$ and $\begin{array}{r}Q\end{array}$ , where the circles overlap , are the points of intersection . Because the radius of each circle is the same, each of these points are equidistant from the endpoints of the line segment. Therefore, each of these points lies on the perpendicular bisector of the line segment.
  ::圆圈重叠的两个点 P 和 Q 是交叉点。 因为每个圆圈的半径相同, 每个圆点与线段端点的等距相等。 因此, 每一个点都位于线段的直角两端。

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  Use your straightedge to draw a line connecting the two intersection points $\begin{array}{r}P\end{array}$ and $\begin{array}{r}Q\end{array}$ . This line bisects the given line segment $\begin{array}{r}AB\end{array}$ , and is called the bisector or, more accurately, the  perpendicular bisector of $\begin{array}{r}AB\end{array}$ .
  ::使用直角绘制一条连接两个交叉点 P 和 Q 的线条。 此线条将给定线段AB 分为两部分, 称为双线段, 或更准确地说, 直线线段的直径。

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  How can you use a compass and straightedge to find the midpoint of a line segment?
  ::您如何使用指南针和直线杆来找到线段的中点 ?

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  One way to find the midpoint of a line segment $\begin{array}{r}AB\end{array}$ is to construct a perpendicular bisector $\begin{array}{r}PQ\end{array}$ . The point $\begin{array}{r}M\end{array}$ , where the perpendicular bisector intersects the line segment, is the midpoint of the line segment.
  ::找到线段AB的中点的一个办法是建立一个垂直双向双向PQ。 点M是线段的中点,即垂直双向双向双向相交的线段。 点M是线段的中点。

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  Bisecting an Angle by Paper Folding
  ::通过纸张折叠比切角

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  Draw an angle like the one below on a piece of paper. Note that the two segments creating the angle do NOT need to be the same length. Bisect the angle by folding the paper.
  ::在一张纸上绘制一个像下面一样的角。 请注意, 创建此角的两个区段不需要相同长度。 通过折叠纸张, 将角度分成两部分 。

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    Fold the paper so that one segment coincides with the other segment. Flatten the paper and crease along the fold.
  ::折叠纸,使一个部分与另一个部分吻合。 折叠纸和折叠折叠。

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  The crease will be the angle bisector .
  ::折痕是两侧角的角

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  Construct an Angle Bisector
  ::构造 角度双扇形

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  Draw an angle to bisect similar to the one below labeled  $\begin{array}{r}\mathrm{\angle }EDF.\end{array}$  Bisect the angle using a compass.
  ::绘制一个与标签为 EDF 的下角相似的角。 使用罗盘将角双切 。

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  Place the compass at the vertex of the angle. Adjust the width of the compass to a medium-wide setting. With $\begin{array}{r}D\end{array}$ as the center , create an arc that passes through both rays , creating an angle marker.
  ::将罗盘放在角度的顶点上。 将罗盘的宽度调整为中等范围设置。 以 D 为中心, 创建一个通过两个射线的弧, 创建一个角标记 。

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  Place the compass on the point $\begin{array}{r}A\end{array}$ , where the arc crosses  $\begin{array}{r}\stackrel{\to }{DE},\end{array}$   and draw a circle . Keeping the width of the compass same, repeat for point $\begin{array}{r}C\end{array}$  on the other  ray , so that the two circles overlap  at two points.
  ::将罗盘放在 A 点上, 弧横过 DEQ, 并绘制一个圆。 将罗盘的宽度保持相同, 在另一射线上重复 C 点, 以便两个圆在两个点上重叠 。

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  The two points where the circles intersect are equidistant  (see image below) from the centers of the small circles, and are therefore equidistant from the  rays creating the angle. Thus , the two points of intersection define the bisector of the angle. Connect those intersection points to create the angle bisector .
  ::圆圈交叉的两点与小圆圈中心相距相当( 见下文图像) , 因此与创造角的射线相距相当。 因此, 两个交叉点可以定义角的两部分 。 连接这些交叉点可以创建角角的两部分 。

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  CK-12 PLIX
  ::CK-12 PLIX(CK-12 PLIX)

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  Constructions and Bisectors
  ::建筑和分部门

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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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  Describe how to construct a $\begin{array}{r}{45}^{\circ }\end{array}$ angle using a compass and a straightedge.
  ::描述如何使用指南针和直角构建一个 45角 。

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  One way to construct a $\begin{array}{r}{45}^{\circ }\end{array}$ angle is to:
  ::建造45角的一个方法就是:

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  • Draw a segment and construct its perpendicular bisector. This will give you $\begin{array}{r}{90}^{\circ }\end{array}$ angles.
    ::绘制一段线段并构造其垂直双向区段。 这将为您提供 90 矩形 。
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  • Construct the angle bisector of one of those $\begin{array}{r}{90}^{\circ }\end{array}$ angles. This will produce two $\begin{array}{r}{45}^{\circ }\end{array}$ angles.
    ::构造其中一个 90 角的角边。 这将产生两个 45 角 。

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  Example 2
  ::例2

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  Prove that the angle bisector created using the method outlined in "Construct an Angle Bisector" above is actually the angle bisector.
  ::证明使用上文“构建一个角两区”中概述的方法创建的角两区实际上就是角两区。

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  Consider the construction in the image. Points have been labeled and two additional segments have been drawn.
  ::考虑图像中的构造。 点被贴上标签, 另绘制了两个部分 。

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  In this picture:
  ::在此图中:

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  • $\begin{array}{r}\overline{AD}\cong \overline{CD}\end{array}$ because they are both radii of the same partial circle centered at point $\begin{array}{r}D\end{array}$ .
    ::AD CD 因为它们都是同一部分圆的半径, 以D点为中心。
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  • $\begin{array}{r}\overline{AB}\cong \overline{CB}\end{array}$ because they are both radii of congruent circles centered at $\begin{array}{r}A\end{array}$ and $\begin{array}{r}C\end{array}$ respectively.
    ::AB {CB} 因为他们都是A和C 分别位于A和C的相近圆圈的半径。
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  • $\begin{array}{r}\overline{BD}\cong \overline{BD}\end{array}$ by the reflexive property.
    ::以反射性财产为例
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  • Therefore, $\begin{array}{r}\mathrm{\Delta }ABD\cong \mathrm{\Delta }CBD\end{array}$ by $\begin{array}{r}SSS\cong \end{array}$ .
    ::因此,SSS的《ABDCBD》。
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  • $\begin{array}{r}\mathrm{\angle }ADB\cong \mathrm{\angle }BDC\end{array}$ because they are corresponding parts of .
    ::ADBDC(ADBBDC),因为它们是ADBDC(ADBBDC)的对应部分。

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  Therefore, $\begin{array}{r}\stackrel{\to }{BD}\end{array}$ must be the angle bisector of $\begin{array}{r}\mathrm{\angle }ADC\end{array}$ .
  ::因此,BD 必须是ZADC的分角。

  Summary

  播放段落 To bisect a segment or an angle means to divide it into two congruent parts. ::将一个段或一个角分成两个相容部分。 播放段落 A bisector of a line segment will pass through the midpoint of the line segment. ::一条线段的一个分区将通过线段的中点。 播放段落 A perpendicular bisector of a segment passes through the midpoint of the line segment and is perpendicular to the line segment. ::一个线段的垂直两分形穿过线段的中点,与线段垂直。 播放段落 An angle bisector is a ray that splits an angle into two congruent, smaller angles. ::角对角是将角分裂成两个相近、较小角的射线。

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  Review
  ::审查审查审查审查

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  1. What does it mean to bisect a segment or an angle?
     ::将片段或角度分成两部分意味着什么?
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  2. Describe the steps for finding the perpendicular bisector of a line segment.
     ::说明为寻找直线段的垂直双向分区段而采取的步骤。
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  3. Describe how to use the perpendicular bisector of a line segment to find the midpoint of the line segment.
     ::说明如何使用直线段的垂直双向分形来找到直线段的中点。
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  4. What's the difference between a bisector and a perpendicular bisector? How can you construct a non-perpendicular bisector of a line segment?
     ::双向和垂直双向之间有什么区别?您如何在直线段中建一个非双向双向区?
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  5. Draw a line segment on your paper and construct the perpendicular bisector of the segment.
     ::在您的纸张上绘制线条段, 并构造线段的垂直双向线段 。
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  6. Draw another line segment on your paper and construct the perpendicular bisector of that segment using another method.
     ::在您的纸张上另绘制一条线段, 并使用其他方法构造该线段的垂直双向线段 。
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  7. Draw an angle on your paper and construct the bisector of the angle.
     ::在您的纸张上绘制一个角度, 并构造角度的双向区段 。
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  8. Draw another angle on your paper and construct the bisector of that angle using another method.
     ::在您的纸张上绘制另一个角度, 并使用其它方法构造该角度的双角区段 。
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  9. Use your straightedge to draw a triangle on your paper. Construct the angle bisector of each angle. What is the point where the bisectors intersect called ?
     ::使用直方在纸张上绘制三角形 。 构造每个角的角双角区段 。 双角区段的相交点是多少 ?
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  10. C onstruct the perpendicular bisector of each side of  a triangle. What is the point where the bisectors intersect ?
      ::构造三角形每侧的垂直双向区段。 双向区段相交的点是多少?
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  11. Construct the medians of  a triangle. What is the point where they intersect ?
      ::构造三角形的中位数。它们相交的点是多少?
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  12. Compare and contrast the two methods for finding a bisector: paper folding vs. compass and straightedge.
      ::比较和对比找到一个分部门的两个方法:折叠纸与罗盘和直线。
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  13. Construct a $\begin{array}{r}{45}^{\circ }\end{array}$ angle (look at  Example 1 for help). Then, construct a $\begin{array}{r}{22.5}^{\circ }\end{array}$ angle.
      ::构造 a 45 gangle (请看例 1 帮助) 。 然后, 构造一个 22. 5 gangle 。
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  14. Construct an isosceles right triangle. Hint: Start by creating a right angle by constructing a perpendicular bisector .
      ::构造右三角形。 提示 : 以创建直角为起点, 构造直角的直角三角形。 提示 : 以创建右角为起点, 构造直角的直角三角形 。 @ info: whatsthis
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  15. If possible, extend your construction of an isosceles right triangle to construct a square. Describe your steps.
      ::如果可能的话, 请扩展您建造的以等分三角形为右三角形以构建方形。 请描述您的步骤 。

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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 。