Course: 5.1 线条线段和角的复制件-interactive

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线段和角的复制件

Constructing Geometric Shapes
::构造几何形状

As you have studied math, you have often created drawings . Drawings are a great way to help communicate a visual idea. 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.
::正如你研究数学一样,你常常会创造图画。绘图是帮助传递视觉想法的好方法。构造与绘制图相似,因为它能产生视觉结果。然而,图画往往只是有助于传递想法的粗略草图,而图画则是用来生成准确几何数字的逐步过程。

Constructions take us back over 2000 years to the ancient Greeks, before computers or other advanced technology. Using only the tools of a compass and a straightedge, they discovered how to very accurately copy segments, angles and shapes, how to create perfect regular polygons, and how to create perfect parallel and perpendicular lines . Today, learning constructions is a way to apply your knowledge of geometric principles. You can do constructions by hand, or with dynamic geometry software. In this lesson , the focus is on hand constructions and making copies of segments, angles, and triangles.
::建筑将我们追溯到2000年的古希腊人身上, 在计算机或其他先进技术之前。他们只使用指南针和直线工具, 发现如何非常准确地复制区块、角度和形状, 如何创建完美的常规多边形, 以及如何创建完美的平行线和直径线。今天, 学习建筑是应用您对几何原理的知识的一种方法。您可以用手动或动态几何软件来进行建筑。在这个教训中, 重点是手动构造, 复制区块、角度和三角形。

To create a construction by hand, there are a few tools that you can use:
::要亲手构建一个构建, 您可以使用一些工具 :

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。

Straightedge: Anything that allows you to produce a straight line . In Geometry construction, a straightedge is actually not used to measure distances. An index card works well as a straightedge. You can also use a ruler as a straightedge . Don't get in the habit of just measuring with the ruler though! Correctly-applied construction steps will result in a more accurate image than estimating b y measuring will.

::直线 : 任何允许您生成直线的事物。在几何构造中, 直线实际上不是用来测量距离的。索引卡和直线工作。您也可以使用标尺作为直线。不要习惯用标尺来测量! 正确应用的构造步骤将产生比用测算意志来估计更准确的图像。

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。

Construct a Copy of a Line Segment
::构造线段的复制件

1 . Draw a line segment on paper with a straightedge. Label it (here we used $\begin{align*}AB).\end{align*}$
::1. 在纸上画一条直线条,并贴上标签(我们在这里使用了AB)。

2. Create a new ray using a straightedge and a pencil, and label it (we used $\begin{align*}MN).\end{align*}$
::2. 使用直线和铅笔建立新的射线,并贴上标签(我们使用了MN)。

3. Measure the length of the original line segment $\begin{align*}AB\end{align*}$ using your compass.
::3. 使用指南针测量原始线段AB的长度。

4. Keeping your compass width the same, place it on the endpoint $\begin{align*}M\end{align*}$ of the ray and draw a little arc with the compass to mark where the endpoint $\begin{align*}P\end{align*}$ should go.
::4. 使罗盘宽度保持相同,将其置于射线的M端点上,并用罗盘绘制一个小弧,以标出P端点应该到哪里去。

5. You have now copied the line segment $\begin{align*}AB\end{align*}$ exactly. The two line segments are congruent, i.e. $\begin{align*}AB = MP\end{align*}$ . You may verify your construction with a ruler, if you wish.
::5. 您现在完全复制了线段 AB 。两个线段是相同的, 即 AB=MP 。如果您愿意, 您可以用标尺来验证您的构造。

Note that in this construction, the compass was used to copy a distance . This is one of the primary uses of a compass in constructions.
::请注意在此构造中, 指南针被用来复制距离。这是在构造中使用指南针的主要用途之一。

Construct a Copy of an Angle
::构造角的复制件

1. Draw an angle that you want to copy on a piece of paper, and label it (we used label $\begin{align*}AOB).\end{align*}$ Keep in mind that what defines the angle is the opening between the two rays . The lengths of the rays are not relevant.
::1. 绘制您想要复制到一张纸上并贴上标签的角(我们使用了 AOB 标签) 。请注意, 角度的定义是两个射线之间的开口。射线的长度无关紧要。

2. Draw a new ray , and label it (we used label $\begin{align*}MN\end{align*}$ below). This ray will become one side of the angle and its end point $\begin{align*}M\end{align*}$ will become the vertex of the angle. Place the compass on the vertex of the angle that you are copying, in this case $\begin{align*}O.\end{align*}$ You can have your compass set to any width that you want to work with. Draw an arc which creates the points $\begin{align*}D\end{align*}$ and $\begin{align*}E\end{align*}$ .
::2. 绘制新的射线并贴上标签(我们用的是标签 MNBEBlow),该射线将成为角度的一面,其终点M将成为角度的顶点。将罗盘放在您正在复制的角度的顶点上,在此情况下,O。您可以将指南针设置在您想要工作的宽度上。绘制一个弧,以创建点D和E。

3. Keeping your compass width the same, place it on endpoint $\begin{align*}M\end{align*}$ of the ray and draw a similar arc . This arc will mark a point where it intersects the ray, identified in the interactive image as point $\begin{align*}P\end{align*}$ .
::3. 使罗盘宽度保持相同,将其置于射线的M端点上,并绘制类似的弧。此弧将标记一个点,将光线交叉,在互动图像中被识别为P点。

4. Use the compass to measure the distance between the two points $\begin{align*}D\end{align*}$ and $\begin{align*}E\end{align*}$ that were created when you drew the arc on the original angle $\begin{align*}AOB.\end{align*}$
::4. 使用指南针测量在原始角ABB上绘制弧时产生的两个D点和E点之间的距离。

5. Without changing the compass width, move the compass to the point $\begin{align*}P\end{align*}$ that you made when your arc crossed the ray. Draw a second arc . This arc will intersect the first arc at point $\begin{align*}Q\end{align*}$ .
::5. 在不改变罗盘宽度的情况下,将罗盘移动到弧过射线时的P点。绘制第二个弧。此弧将交叉到Q点的第一个弧。

6. Use a straightedge to draw another ray (here labeled $\begin{align*}MR\end{align*}$ ) from the vertex $\begin{align*}M\end{align*}$ passing through the point of intersection $\begin{align*}Q\end{align*}$ . You have now copied the angle $\begin{align*}AOB.\end{align*}$ T he two angles are congruent , i.e. $\begin{align*}\angle AOB = \angle NMR\end{align*}$ . You may verify your construction with a protractor .
::6. 使用直角从通过交叉点Q的顶部M线上再画一个射线(这里贴有MR标签)。您现在复制了角AOB, 两种角度是相同的, 即 AOBNMR。您可以用一个减压器来验证您的构造。

Construct an Exact Copy of an Angle
::构造一个角的精确副本

1. Start by copying the line segment on the bottom, using the process outlined in Construct a Copy of a Line Segment (draw a ray, use the compass to measure the width of the line segment, mark off the endpoint on the ray). Construct a copy of the angle using the process outlined in Construct a Copy of an Angle (draw an arc through the angle and draw the same arc through the new ray, measure the width of the arc, draw a new ray through the intersection of the two markings).
::1. 通过在底部复制线段开始,使用“构造线段复制件”(绘制一线,使用罗盘测量线段的宽度,在射线上标记端点)中描述的过程。使用“构造角复制件”中描述的过程(通过角度绘制弧,通过新的射线绘制相同的弧,测量弧宽度,通过两个标记的交叉点绘制新的射线)中描述的角复制件。

2. Finally, copy the second line segment by measuring its length using the compass and marking off the correct spot for the endpoint. You have now copied the exact angle (including the additional points marked $\begin{align*}A \text{ and } B\end{align*}$ on the original angle). The two angles are congruent. You may v erify your construction with a protractor.
::2. 最后,复制第二行段,使用罗盘测量其长度,并在端点的正确位置上打上标记。您现在复制了确切角度(包括原角上标记为A和B的额外点),这两个角度是相同的。您可以用一个减速器来验证您的构造。

Examples
::实例实例实例实例

To copy a triangle means to create a congruent triangle. There are four sets of triangle congruence criteria that work for any type of triangle: , , AAS, and ASA . You can use SSS, SAS, or ASA, combined with the process of copying angles and line segments to copy a triangle.
::要复制三角形, 则要创建一致的三角形。对于任何类型的三角形, 有四套三角形一致的标准: 、、 AAS 和 ASA 。您可以使用 SSS、 SAS 或 ASA, 结合复制角度和线条段的过程来复制三角形。

To try the following examples yourself, use a straightedge to draw a triangle on your paper. Use the processes described in the examples to copy the triangle in different ways.
::要尝试下面的例子, 请使用直方在您的纸张上绘制三角形。使用示例中描述的流程以不同方式复制三角形。

Example 1
::例1

Copy the given triangle using SSS.
::使用 SSS 复制给定三角形。

Create a new ray (marked as $\begin{align*}DN\end{align*}$ in the interactive image below).
::创建新射线( 在下面交互式图像中标记为 DN ) 。

Use your compass to measure the length of side $\begin{align*}AB\end{align*}$ of the given triangle .
::使用您的指南针测量给定三角形的侧边 AB 长度。

Keeping your compass width the same, draw a little arc to mark the other vertex $\begin{align*}E\end{align*}$ .
::将罗盘宽度保持相同, 绘制一个小弧以标记另一个顶点 E 。

Use the compass to measure the length of side $\begin{align*}BC\end{align*}$ .
::使用指南针测量 BC 侧边长度。

Keeping the compass width the same, place it on vertex $\begin{align*}E\end{align*}$ and draw an arc to mark the length of the second side of the triangle.
::将罗盘宽度保持相同, 将其放在顶端 E 上, 并绘制弧以标记三角形第二侧的长度。

U se the compass to measure the length of side $\begin{align*}AC\end{align*}$ .
::使用指南针测量侧向 AC 的长度。

Without changing the compass width, place it on vertex $\begin{align*}D\end{align*}$ , and draw an arc intersecting the previous one to get the third vertex $\begin{align*}F\end{align*}$ of the triangle.
::在不更改罗盘宽度的情况下, 将其置于顶点 D 上, 并绘制一个弧交叉, 以获得三角形的第三个顶点 F 。

Connect to form the triangle $\begin{align*}DEF.\end{align*}$
::连接以形成三角DEF。

You have now copied the triangle $\begin{align*}ABC\end{align*}$ exactly.
::您现在完全复制了三角形ABC。

Note that with this method, you have only used the lengths of the sides of the triangle (as opposed to any angles) to construct the new triangle.
::请注意,使用此方法,您只使用三角形边的长度(相对于任何角度)来构造新三角形。

Example 2
::例2

Copy the given triangle using SAS.
::使用 SAS 复制给定三角形。

Create a new ray $\begin{align*}DN.\end{align*}$
::创建新射线 DN 。

Measure the length of side $\begin{align*}AB\end{align*}$ of the given triangle using your compass .
::使用您的罗盘测量给定三角形的侧边 AB 长度。

K eeping your compass width unchanged , draw a little arc to mark the other vertex $\begin{align*}E\end{align*}$ .
::保持您的罗盘宽度不变, 绘制一个小弧以标记另一个顶点 E 。

Copy angle $\begin{align*}ABC:\end{align*}$
- Draw an arc through the original angle.
  ::通过原始角度绘制弧。
- Draw the same arc through the new ray.
  ::通过新射线绘制相同的弧。
- Measure the width of the arc.
  ::测量弧的宽度。
- Draw an arc intersecting the previous one.
  ::绘制前一个相交的弧。
- Draw a new ray through the intersection of the two markings.
  ::通过两个标记的交叉点绘制新的射线。
::复制角度 ABC: 通过原始角度绘制弧。通过新射线绘制相同的弧。测量弧的宽度。绘制前一个弧的交叉点。通过两个标记的交叉点绘制新的射线。

Copy the second side $\begin{align*}BC\end{align*}$ by measuring its length using your compass and marking off the correct spot for the vertex $\begin{align*}F\end{align*}$ onto the ray that you just drew.
::复制 BC 第二面, 使用您的指南针测量其长度, 并在您刚刚绘制的射线上标记 F 顶点的正确位置。

Connect $\begin{align*}DF\end{align*}$ to form the triangle.
::连接 DF 以形成三角形。

You have now copied triangle $\begin{align*}ABC\end{align*}$ exactly.
::你现在完全复制了三角形ABC

Example 3
::例3

Copy the given triangle using ASA.
::使用 ASA 复制给定三角形。

Create a new ray $\begin{align*}DN.\end{align*}$
::创建新 RayDN 。

Measure the width of the side $\begin{align*}AB\end{align*}$ of the given triangle using your compass .
::使用您的罗盘测量给定三角形的侧边 AB 宽度。

K eeping your compass width unchanged , draw a little arc to mark the other vertex $\begin{align*}E\end{align*}$ .
::保持您的罗盘宽度不变, 绘制一个小弧以标记另一个顶点 E 。

Copy the angle $\begin{align*}ABC:\end{align*}$
- Draw an arc through the original angle .
  ::通过原始角度绘制弧。
- Then draw the same arc through the new ray, and measure the width of the arc.
  ::然后通过新射线绘制相同的弧,测量弧的宽度。
- Finally , draw an arc intersecting the previous one and draw a new ray through the intersection of the two markings.
  ::最后,绘制一个交叉弧,将上一个横线交叉,并用新的射线通过两个标记的交叉线。
::复制角度 ABC : 在原始角度中绘制弧。然后通过新射线绘制相同的弧, 并测量弧的宽度。最后, 绘制一个弧交叉, 并绘制一个新的射线, 通过两个标记的交叉点。

Copy the angle at the other endpoint $\begin{align*}D\end{align*}$ .
::在另一端点D复制角度。

The point of intersection of the two rays is marked as the vertex $\begin{align*}F\end{align*}$ .
::两个射线的交叉点标记为顶点F。

Connect $\begin{align*}DF\end{align*}$ to form the triangle.
::连接 DF 以形成三角形。

You have now copied the triangle $\begin{align*}ABC\end{align*}$ exactly.
::您现在完全复制了三角形ABC。

Summary

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。

Straightedge : Anything that allows you to produce a straight line.
::直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线直线

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。

An intersection is the point or set of points where lines, planes, segments, or rays cross each other.
::交叉点是指线条、平面、片段或射线相互交错的点或一组点。

A line segment is a finite "part" of a line. Line segments have two endpoints.
::线条段是线条中有限的“部分”。线条段有两个端点。

You can use SSS, SAS, or ASA, combined with the process of copying angles and line segments to copy a triangle.
::您可以使用 SSS、SAS 或 ASA, 结合复制角度和线段的过程来复制三角形。

Review
::审查审查审查审查

What is the difference between a drawing and a construction?
::画画和建筑有什么区别?

What is the difference between a straightedge and a ruler?
::直立与统治者有什么区别?

Describe the steps for copying a line segment.
::描述复制线段的步骤。

Describe the steps for copying an angle.
::描述复制角度的步骤。

When copying an angle, do the lengths of the lines matter? Explain.
::复制角度时, 行的长度重要吗 ? 解释。

Explain the connections between copying a triangle and the triangle congruence criteria.
::解释复制三角形与三角形一致标准之间的关联。

Draw a line segment and copy it with a compass and straightedge.
::绘制一条线段, 并用罗盘和直线复制它。

Draw another line segment and copy it with a compass and straightedge.
::绘制另一条线段, 并用指南针和直线复制它。

Draw an angle and copy it with a compass and straightedge.
::绘制角度, 并用指南针和直角复制它。

Draw another angle and copy it with a compass and straightedge.
::绘制另一个角度, 并用指南针和直角复制它。

Use your straightedge to draw a triangle. Copy the triangle using $\begin{align*}SSS \cong\end{align*}$ . Describe your steps.
::使用直方键绘制三角形。使用 SSS Q\\ 复制三角形。描述您的步骤。

Copy the triangle from #11 using $\begin{align*}SAS \cong\end{align*}$ . Describe your steps.
::使用 SAS * 复制从 # 11 复制的三角形。描述您的步骤。

Copy the triangle from #11 using $\begin{align*}ASA \cong\end{align*}$ . Describe your steps.
::使用 ASA\\\\\\ 复制从 # 11 复制的三角形。描述您的步骤。

Graph the points A(1, 4), B(4, 2) and C(1, 5). Join the points to create line segment AC. Do you think it is possible to find the slope of this line? Explain.
::图形 A( 1 , 4) 、 B( 4, 2) 和 C( 1, 5) 。加入这些点以创建线段 AC。您认为可以找到这条线的斜度吗? 解释。

Draw a triangle. Describe the differences in your drawing if you were to show ASA, SAS, or SSS.
::绘制三角形。如果显示 ASA、 SAS 或 SSS ,请在绘图中描述差异。

Do you think a compass is more or less accurate than a ruler? What are the advantages of using a compass when constructing a triangle?
::您认为指南针比标尺更准确吗? 在构建三角形时使用指南针有什么好处?

Review (Answers)
::审查(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。

Section outline

General

线段和角的复制件

Constructing Geometric Shapes ::构造几何形状

Construct a Copy of a Line Segment ::构造线段的复制件

Construct a Copy of an Angle ::构造角的复制件

Construct an Exact Copy of an Angle ::构造一个角的精确副本

Examples ::实例实例实例实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::审查审查审查审查

Review (Answers) ::审查(答复)