Course: 9.3 圆形中的弧 | The Way To Learn

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圆形中的弧
Arcs in Circles
::圆形中的弧

A circle has $\begin{aligned} 360^{\circ} \end{aligned}$ . An arc is a section of the circle. A semicircle is an arc that measures $\begin{aligned} 180^{\circ} \end{aligned}$ .
::圆有360。弧是圆的一个部分。半圆是测量180的弧。

$\begin{aligned} \hat{E H G} and \hat{E J G} are semicircles \end{aligned}$
::EHG和EJG是半圆形

A central angle is the angle formed by two radii with its vertex at the center of the circle. A minor arc is an arc that is less than $\begin{aligned} 180^{\circ} \end{aligned}$ . A major arc is an arc that is greater than $\begin{aligned} 180^{\circ} \end{aligned}$ . Always use 3 letters to label a major arc.
::中心角是两个半径形成的角度, 其顶点在圆圈的中心。次要弧是小于180的弧。主要弧是大于180的弧。总是用三个字母来标出一个大的弧。

The central angle is $\begin{aligned} ∠ B A C \end{aligned}$ . The minor arc is $\begin{aligned} \hat{B C} \end{aligned}$ . The major arc is $\begin{aligned} \hat{B D C} \end{aligned}$ .
::中心角是 BAC。次要弧是 BCQ。主要的弧是 BDCQ。

An arc can be measured in degrees or in a linear measure (cm, ft, etc.). In this concept we will use degree measure. The measure of a minor arc is the same as the measure of the central angle that corresponds to it . The measure of a major arc is $\begin{aligned} 360^{\circ} \end{aligned}$ minus the measure of the corresponding minor arc. The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs ( Arc Addition Postulate ).
::弧可以以度或线性测量( cm, ft等) 。在这个概念中, 我们将使用度量。小弧的度量与与其对应的中心角的度量相同。大弧的度量是 360 减去相应的次要弧的度量。由两个相邻弧组成的弧的量量是两个弧的量的和( 弧附加假设)。

$\begin{aligned} m \hat{A D} + m \hat{D B} = m \hat{A D B} \end{aligned}$

::亚马逊河流域

What if a circle were divided into pieces by various radii? How could you find the measures of the arcs formed by these radii?
::如果圆被各种弧形分割成碎片呢?你如何找到这些弧形形成的弧体的量度?

Examples
::实例

Example 1
::例1

Find $\begin{aligned} m \hat{A B} \end{aligned}$ and $\begin{aligned} m \hat{A D B} \end{aligned}$ in $\begin{aligned} ⨀ C \end{aligned}$ .
::寻找 mAB和mADB 在C。

$\begin{aligned} m \hat{A B} = m ∠ A C B \end{aligned}$ . So, $\begin{aligned} m \hat{A B} = 102^{\circ} \end{aligned}$ .
::mABmACB。所以,mAB102。

$\begin{aligned} m \hat{A D B} = 360^{\circ} - m \hat{A B} = 360^{\circ} - 102^{\circ} = 258^{\circ} \end{aligned}$

::360360102258

Example 2
::例2

Find the measures of the minor arcs in $\begin{aligned} ⨀ A \end{aligned}$ . $\begin{aligned} \bar{E B} \end{aligned}$ is a diameter .
::在 A.EB 中找到小弧的量度。 EB 是直径。

Because $\begin{aligned} \bar{E B} \end{aligned}$ is a diameter, $\begin{aligned} m ∠ E A B = 180^{\circ} \end{aligned}$ . Each arc has the same measure as its corresponding central angle.
::因为 EB 是直径, mEAB=180 。每个弧都有与其相应的中心角相同的量度。

$\begin{aligned} m \hat{B F} & = m ∠ F A B = 60^{\circ} \\ m \hat{E F} & = m ∠ E A F = 120^{\circ} \to 180^{\circ} - 60^{\circ} \\ m \hat{E D} & = m ∠ E A D = 38^{\circ} \to 180^{\circ} - 90^{\circ} - 52^{\circ} \\ m \hat{D C} & = m ∠ D A C = 90^{\circ} \\ m \hat{B C} & = m ∠ B A C = 52^{\circ} \end{aligned}$

::mBFmmmmmeAF=12018060mEDmED=38 18090 52mDCmDAC=90mBCmBAC=52

Example 3
::例3

Find the measures of the indicated arcs in $\begin{aligned} ⨀ A \end{aligned}$ . $\begin{aligned} \bar{E B} \end{aligned}$ is a diameter.
::查找在 A. EB 中显示的弧的量度。 EB 是直径。

Use the Arc Addition Postulate.
::使用Arc 附加假设。

$\begin{aligned} m \hat{F E D} \end{aligned}$
::mFED

$\begin{aligned} m \hat{F E D} = m \hat{F E} + m \hat{E D} = 120^{\circ} + 38^{\circ} = 158^{\circ} \end{aligned}$
::mFED * mFED * 120 * 38 * 158 * mFED * mFED * mFED * mFED * mED * 120 * 38 * 158 * mFED * mFED * mFED * mFED * mFED * mFED * mED * 120 * 38 * 158 *158 *$158 *

$\begin{aligned} m \hat{C D F} \end{aligned}$
::mCDF__________________________________________________________________________________

$\begin{aligned} m \hat{C D F} = m \hat{C D} + m \hat{D E} + m \hat{E F} = 90^{\circ} + 38^{\circ} + 120^{\circ} = 248^{\circ} \end{aligned}$
::mCDFmCDmDEmEF9038120248

$\begin{aligned} m \hat{D F C} \end{aligned}$
::mDFC

$\begin{aligned} m \hat{D F C} = m \hat{E D} + m \hat{E F} + m \hat{F B} + m \hat{B C} = 38^{\circ} + 120^{\circ} + 60^{\circ} + 52^{\circ} = 270^{\circ} \end{aligned}$
::mDFCmEFmFBMB381206052270

Example 4
::例4

List the congruent arcs in $\begin{aligned} ⨀ C \end{aligned}$ below. $\begin{aligned} \bar{A B} \end{aligned}$ and $\begin{aligned} \bar{D E} \end{aligned}$ are diameters.
::列出以下 C 中的一致弧。 AB 和 DE 是直径。

$\begin{aligned} ∠ A C D ≅ ∠ E C B \end{aligned}$ because they are . $\begin{aligned} ∠ D C B ≅ ∠ A C E \end{aligned}$ because they are also vertical angles.
::因为他们也是垂直角度。

$\begin{aligned} \hat{A D} ≅ \hat{E B} \end{aligned}$ and $\begin{aligned} \hat{A E} ≅ \hat{D B} \end{aligned}$
::ADEB 和 AEDB 和 AEDB 和 AEDB

Example 5
::例5

For each of the circles below, are the blue arcs congruent? Explain why or why not.
::对于下面的每个圆圈来说,蓝色弧是否一致?解释为什么或为什么不是。

For the first circle, $\begin{aligned} \hat{A D} ≅ \hat{B C} \end{aligned}$ because they have the same central angle measure and are in the same circle.
::对于第一个圆,ADBC,因为它们具有相同的中心角度量,处于同一个圆。

For the second circle, the two arcs have the same measure, but are not congruent because the circles have different radii.
::对于第二个圆,两个弧具有相同的度量,但并不一致,因为圆圈的弧度不同。

Review
::回顾

Determine whether the arcs below are a minor arc, major arc, or semicircle of $\begin{aligned} ⨀ G \end{aligned}$ . $\begin{aligned} \bar{E B} \end{aligned}$ is a diameter.
::确定下面的弧是小弧、大弧还是G的半圆圈。 EB是直径。

$\begin{aligned} \hat{A B} \end{aligned}$
::AB_______________________________________________________________________________________________________________________________

$\begin{aligned} \hat{A B D} \end{aligned}$
::ABD_____________________________________________________________________________________________________________________

$\begin{aligned} \hat{B C E} \end{aligned}$
::BCE_______________________________________________________________________________________

$\begin{aligned} \hat{C A E} \end{aligned}$
::开开开

$\begin{aligned} \hat{A B C} \end{aligned}$
::ABC( ABC)

$\begin{aligned} \hat{E A B} \end{aligned}$
::预事项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项

Are there any congruent minor arcs? If so, list them.
::是否有一致的小弧? 如果是的话,请列出来。

If $\begin{aligned} m \hat{B C} = 48^{\circ} \end{aligned}$ , find $\begin{aligned} m \hat{C D} \end{aligned}$ .
::如果 mBC48, 找到mCD。

Using #8, find $\begin{aligned} m \hat{C A E} \end{aligned}$ .
::使用#8,找到 MCAE。

Find the measure of the minor arc and the major arc in each circle below.
::在以下每个圆圈中查找小弧和大弧的量度。

Determine if the blue arcs are congruent. If so, state why.
::确定蓝色弧是否相容。如果是, 请说明原因。

Find the measure of the indicated arcs or central angles in $\begin{aligned} ⨀ A \end{aligned}$ . $\begin{aligned} \bar{D G} \end{aligned}$ is a diameter.
::在 A. DG 中查找指示弧或中心角的量度。 DG 是直径。

$\begin{aligned} \hat{D E} \end{aligned}$
::德德德

$\begin{aligned} \hat{D C} \end{aligned}$
::DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_ DC_

$\begin{aligned} \hat{G A B} \end{aligned}$
::性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别

$\begin{aligned} \hat{F G} \end{aligned}$
::财务部

$\begin{aligned} \hat{E D B} \end{aligned}$
::EDB_______________________________________________________________________________________________________________________________

$\begin{aligned} \hat{E A B} \end{aligned}$
::预事项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项项

$\begin{aligned} \hat{D C F} \end{aligned}$
::DCF_______________________________________________________________________________________

$\begin{aligned} \hat{D B E} \end{aligned}$
::DBE

Find the measure of $\begin{aligned} x \end{aligned}$ in $\begin{aligned} ⨀ P \end{aligned}$ .
::在 QP 中查找 x 的量度。

Review (Answers)
::回顾(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

圆形中的弧

Arcs in Circles ::圆形中的弧

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Arcs in Circles
::圆形中的弧

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)