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专题五圆中综合应用讲义

板块一：辅助圆——图中无圆，心中有圆

【热身】已知：如图，四边形ABCD 中，BD 平分∠ABC ，∠A 、∠C 互补.求证：AD =CD

【引入】如图，Rt △ABC 中，∠C =90°，∠ABC =30°，AB =6，点D 在AB 边上，点E 是

BC 边上一点（不与点B ，C 重合），且DA =DE ，则AD 的取值范围是．

已知△ABC 中，∠A =45°，BC =3，AC =a ，若满足上述条件的△ABC 有且只有一个，则a 的取值范围为______.

在△ABC 中，AB =AC =2，BC 边上有100个不同的点P 1，P 2，……，P 100。记

()212,,100i i i i m A P B P P C i =+?=

L ，，则12100...m m m +++= .

在坐标平面内，与点A （1，2）距离为1，且与点B （3，1）距离为2的直线共有（）

A ．1条

B ．2条

C ．3条

D ．4条

板块二：圆中角的灵活转化

补充定理1：圆内接四边形定理圆的内接四边形性质定理：

性质定理1：圆内接四边形的对角互补；

性质定理2：圆内接四边形的一个外角等于它的内对角. 圆内接四边形判定定理：

如果一个四边形的对角互补,那么它的四个顶点共圆. 补充定理2：弦切角定理

弦切角：顶点在圆上，一边和圆相交，另一边和圆相切的角叫做弦切角.(弦切角就是切线与弦所夹的角）

弦切角定理：弦切角等于它所夹的弧所对的圆周角.

如图，AB 是圆O 的直径，直线EF 切圆O 于点B ，C 、D 是圆O 上的点，弦切角 ∠CBE =40°，AD =CD ，则∠BCD 的度数是（）

A ．110°

B ．115°

C ．120°

D ．135°

如图，已知AC =CB ，∠C =90°，过点C 作直线l //AB ，以A 为圆心，AB 为半径作圆交直线l 于M 、N ，求∠CMB 和∠CNB 的大小.

如图，△ABC 为圆的内接三角形，D 为AB 上一点，且4AD =AB .P 在圆上，

且∠ADP =∠ACB ，若PD PB =______.

板块三：圆中比例线段

补充定理3：相交弦定理

圆内两条相交弦，被交点分成两条线段长的积相等； (经过圆内一点引两条弦，各弦被这点所分成的两段的积相等) 补充定理4：切割线定理

从圆外一点引圆的切线和割线，切线长是这点到割线与圆交点的两条线段长的比例中项．补充定理4：切割线定理推论（割线定理）

从圆外一点引圆的两条割线，这一点到每条割线与圆的交点的两条线段长的积相等．

已知半径为1和2的同心圆1O e 、2O e ，在1O e 的圆周上任取一点P 作两条互相垂直的弦交2O e 于AB

、CD =（）

A ．

B ．

C ．

D ．与弦的位置有关

已知AB 是半径为1的O e 的一条弦，且AB =a <1，以AB 为一边在O e 内作正三角形ABC ，D 为O e 上不同于点A 的一点，且DB =AB =a ，DC 的延长线交O e 于点E ，则AE 的长为（）

．

B ．1 C

． D ．a 如图，PA 切圆O 于A ，割线PBC 交圆O 于B 、C 两点，D 为PC 的中点，连接AD 并延长交圆O 于E ，已知：EA DE BE ?=2

1）求证：PA =PD 2）DE AD BP ?=2

板块四：综合应用

在直角扇形OAB 中，OA =OB =1，在AB 弧上任取一点C ，过C 作CD ⊥OB 于D ，则OD +DC 的最大值为______.

如图所示，⊙O 的直径的长是关于x 的二次方程()0222

=++k x k x -（k 是整数）的最

大整数根，P 是⊙O 外一点，过点P 做⊙O 的切线P A 和割线PBC ，其中A 为切点，点B ，C 是直线PBC 与⊙O 的交点。若PA ，PB ，PC 的长都是正整数，且PB 的长不是合数，求

222PA PB PC ++的值.

2008年西交大少年班英语考试试题

本试卷分第Ⅰ卷（选择题）和第Ⅱ卷（非选择题）两部分。第Ⅰ卷1至7页。第Ⅱ卷8－9页，共100分。考试时间120分钟。注意：第一卷的试题答案请答在答题卡上，答在试卷上一律无效。

第Ⅰ卷（选择题共65分）

第一部分：知识运用（共两节，35分）

第一节单项选择 (共15小题；每小题1分, 共15分)

从A、B、C、D四个选项中，选出可以填入空白处的最佳选项，并在答题卡上将该项涂黑。

1. In the USA, _____ students not only need good grades, but also need to get _____ social experience.

A. the; the

B. a; a

C. /; /

D. the; a

2. Have you rea d the paper today? Train ______ are going up again, and they’re so expensive already!

A. fees

B. fares

C. prices

D. tickets

3. —Thank you for your good advice on how to plan for the future, Mrs. Williams.

—__________.

A. It doesn’t matter

B. With pleasure

C. My pleasure

D. My happiness

4. The latest online survey, _____ by https://www.wendangku.net/doc/155221211.html,, found that more than 73 per cent of young people want to work as civil servants.

A. to be carried out

B. being carried out

C. carried out

D. having been carried out

5. He was robbed in a lonely street last night, but _______ he didn’t have any money or credit cards in his wallet at that time.

A. fortunately

B. happily

C. finally

D. eventually

6. With homeschooling growing quickly in the United States, nobody is quite sure exactly how

many American children _____ at home.

A. being taught

B. taught

C. are being taught

D. are teaching

7. The police suspected him of carrying drugs so they _____ his bag, but they found nothing.

A. went on

B. went in

C. went across

D. went through

8. Abdulla has enjoyed his life in China, but he says that sometimes he wishes Chinese _____

more about his homeland in Africa.

A. had known

B. know

C. knew

D. have known

9. I will take my daughter with me when I _________ ShangHai.

A. go to

B. will go to

C. have been to

D. have gone to

10. When I saw Mary, she ________ the violin in the sitting room.

A. is playing

B. plays

C. was playing

D. played

11. I didn't want either of _______ shirts and asked the salesman to show me_________.

A. those, another

B. two, the other

C. all, the others

D. that, others

12. _____________ wants the book may have it.

A. Who

B. Whoever

C. Anyone

D. The person

13. Is this the museum ___________ your brother worked 2 years ago?

A. that

B. which

C. where

D. the one

14. I was doing my homework __________ he was playing the computer game.

A. when

B. while

C. before

D. as

15. She had escaped, __________the ring had fallen off and been damaged in the great heat of the fire.

A. so

B. or

C. but

D. and

第二节完形填空 (共20小题；每小题1分，共20分)

阅读下面短文，掌握其大意，从每题所给的A、B、C、D四个选项中，选出最佳选项，并在答题卡上将该项涂黑。

On the last day before Christmas, I hurried to the supermarket to buy the remaining gifts I hadn’t managed to get earlier. When I saw all the 16 there, I started to say to my self: “It’s going to 17forever here and I still have so many other places to go. Christmas really is getting more and more 18every year. How I wish I could just lie down, go to sleep and only wake up 19 it’s over…”

Nevertheless, I made my way to the toy 20 , and there, I started to complain of the 21 as I wondered if kids really played 22 such expensive toys. While looking around the 23 , I noticed a boy of about five, pressing a 24 against his chest.

He kept 25 the hair of the doll and looked quite 26 . I wondered who the doll was for. Then the boy 27 an old woman beside him.

“Granny, are you sure I don’t have enough 28 ?” She replied, “You know that you don’t have enough to buy this doll, my dear.”

Then she asked him to stay there for five minutes 29 she looked around. She left quickly and the boy continued to 30 the doll in his hand. I walked towards him and asked who he 31 to give the doll to.

“It is the doll my 32 loved most and wanted so much for Christmas. She was so 33 that Santa Claus would bring it to her.”

I told him that maybe Santa Claus would bring it after all, and not to 34 , but he said sadly, “No, Santa Claus cannot take it where she is now. My sister has gone to be with 35 . Daddy says mummy will also go to see God very soon, so I thought that she could take the doll with her to give my sister.”

16. A. gifts B. people C. goods D. children

17. A. take B. stay C. remain D. delay

18. A. pleasant B. discouraging C. annoying D.exciting

19. A. before B. instantly C. immediately D. after

20. A. part B. section C. area D. group

21. A. quality B. sizes C. prices D. colour

22. A. with B. at C. on D. about

23. A. room B. shelves C. crowd D. toys

24. A. toy B. hand C. doll D. chin

25. A. seeing B. touching C. pulling D. taking

26. A. angry B. happy C. excited D. sad

27. A. turned to B. ran to C. shouted at D. called at

28. A. time B. money C. energy D. power

29. A. so that B. while C. until D. the moment

30. A. take B. catch C. hold D. stick

31. A. wished B. asked C. supposed D. believed

32. A. friend B. mummy C. granny D. sister

33. A. painful B. worried C. sure D. moved

34. A. take B. worry C. leave D. move

35. A. mummy B. God C. daddy D. grandma

第二部分：阅读理解（共20小题；每小题1.5分，共30分）

阅读下列短文，从每题所给的A、B、C、D四个选项中，选出最佳选项，在答题卡上将该项涂黑。

Maybe your class in school has given a play. People throughout the world like to act in plays. In Japan, actors perform in Kabuki plays. The word “Kabuki” is made up of three Japa nese words meaning song, dance and ability.

Kabuki actors do not look like the actors in American plays. American actors dress and look like real people. In Kabuki plays, the actors wear bright-colored robes and wigs. Their robes are very large, and their wigs do not look like real hair.

American actors wear make-up, but their make-up does not often hide their faces. Kabuki actors paint their faces chalk white. They draw black eyebrows above their real eyebrows. They outline their eyes in black or red. Their mouths are bright red. They look as if they are wearing masks.

An actor performing in an American play must make his face look happy, sad or angry. Make–up helps the Kabuki actor show his feelings. If an actor is going to show anger, he paints dark blue or red lines on his face. His make-up makes him look angry.

American and Kabuki actors perform in different ways. But they both try to please the people who watch them.

36. The underlined word “wigs”in the second paragraph probably means _________.

A. make-up

B. clothes

C. masks

D. articifical hair

37. From the story we may know that .

A. American actors don’t wear make-up

B. Kabuki actors don’t want to show their real faces

C. actors in America and Japan both hope to make audience happy and relaxed

D. American performances are much better than Japanese ones

38. To show anger, a Kabuki actor paints .

A. dark blue or red lines on his face

B. hair on his head

C. his costume blue

D. his eyebrows black

39. The text is mainly about .

A. how the Japanese sing and dance

B. why actors look sad or happy

C. a kind of Japanese play

D. make-up in plays

40. Kabuki actors wear so much make-up because __________.

A. they do not want anyone to know who they are

B. it helps them show their feelings in a

play

C. they think it makes them prettier

D. they want to cover up their true

feelings

In spring, the first images that come into mind are of an outing for a picnic or travel. Regardless of what spring has in store for you, you will probably need a bike to do it. If you don't know how to ride one, this week's shopping page will be a little useless for you. How sad, you missed out on one of the best memories of youth. More and more people are interested in building your own bike and choosing its color, appearance, function and accessories. This week, we're showcasing parts:

Frame

Making your own bike is not cheap! If you start with this frame, your bill will already be up to 260 yuan - almost the cost of an assembled bicycle. This 21 frame is unmarked and comes in two colors, white and yellow. At 1890g, it's one of the most functional in town.

Available: https://www.wendangku.net/doc/155221211.html, Price: 260 yuan

Winzip disc-brake

Winzip? No, it's not the archiving software on your computer. This Winzip is a bike parts company. This disc-brake has 160 teeth and costs 250 yuan. Anyone who specializes in cycles can tell you how great disc brakes are. Of course, don't take word for it. Give them a try!

Available: Giant, No 3, Yuetan Bei Jie, Xicheng Price: 250 yuan

KMC-X10 10 speed chain

This KMC chain looks like something some kind of metal monster that might make a snack out of you. If you use this one though, expect a silent ride. A good chain can save you a lot of energy when peddling.

Available: No 49, Bei Sanhuan Zhong Lu, Chaoyang Price: 65 yuan

Alexrims Wheel

"I like to rest my head on the handlebars to listen to the wheels while they crunch the grass and leaves," one boy said. What a lovely image! If you don't want to ruin the scene, buy a set of Alexrims wheels. At 500g, with 6061-T6 material, they will be extremely comfortable. It's worth noting that Alexrims is a top brand.

Available: Armini Bicycle, No 91, Xisi Nan Dajie, Xicheng Price: 160 yuan each

Truvativ-XR handlebars

Handlebars are where you can rest your hands, head, or any other body part. Strike a pose by grabbing your handlebars and pretend you actually know how to ride. Wow! You'll be popular with the girls in no time.

Available: Giant, 101, Building E, Guohengjiye Building, No 7, Bei Tucheng Xi Lu,

Chaoyang Price: 280 yuan

41. This passage is an advertisement for selling ______.

A. computers

B. a software on the computer

C. bikes

D. parts of bikes

42. Which of the following are sold in stores located in the district of Chaoyang?

A. Winzip disc-brake and KMC-X10 10 speed chain

B. Truvativ-XR handlebars and Alexrims Wheel

C. Winzip disc-brake and Alexrims Wheel

D. KMC-X10 10 speed chain and Truvativ-XR handlebars

43. This advertisement is intended to be read by_____________.

A. the parents

B. the elderly

C. the youth

D. the technicians

44. According to the advertiser, _________________.

A. this week’s shopping page will teach you how to ride a bike

B. building a bicycle is more expensive than an assembled one

C. Alexrims Wheels can make loud noises

D. driving a car is more attractive for girls than riding a bicycle

45. Which of the following is the best title of the ad?

A. Enjoy Spring on Two Wheels

B. Spring Is Coming

C. Colorful Bikes Are on Sale

D. Does Your Bikes Need Repairing?

One day last year, there was a sudden knock on the door. Without warning, my nephew had arrived from Turkey! When I had last seen him, he was knee-high to a grasshopper（蚂蚱）, with shy eyes, ears like two fans, two front teeth missing, short hair and continually dirty hands. You know, the look that fits every nephew. I liked and was closely attached to him. With that

knee-high-to-a-grasshopper size, he used to look up at me as if viewing a telephone pole, his blue eyes smiling and secretly making fun of me. The legs sticking out of his short pants were a little bent. Through his eyes were straight, he appeared a bit cross-eyed. I felt sorry when I looked at him. When we talked, he seemed to have a weight on his shoulders and appeared offended. When he was guilty, his eyes grew wet and his voice softened to where he could hardly be heard. Those who saw him thought him an orphan（孤儿）and felt sorry. They felt like putting their hands in their pockets and giving him some spending money or candy. In spite of my hitting my other nephews for any old thing, this one I couldn’t touch. I loved the little son of a gun!

At home, no matter who got angry, our nephew managed to keep his distance. If you spoke to him, he didn’t reply. If he did answer, it was quietly. Even if you hit him, he was quiet. When taking a beating, instead of increasing, his wailing（嚎啕哭号）decreased. Thus, the anger of whoever was beating him turned to pity and the boy was saved from further punishment. When talking with others, I observed that our boy had neither crooked legs, cross-eyes nor big ears hanging like leaves. Furthermore, when he got mad, he knew how to shout his head off. It was only when he detected danger that his legs went crooked, his ears grew and his eyes crossed.

46. The author loved the nephew he described because __________.

A. the nephew was an orphan

B. the author loved all of his nephews

C. the nephew looked nice and lovely

D. the nephew’s appearance called forth his

love

47. When the boy was beaten at home, ___________.

A. he shouted his head off

B. his decreased wailing made the person beating him show pity on him

C. he tried to run away to escape punishment

D. his eyes grew wet and his voice softened to where he could hardly be heard

48. Why did the author describe his nephew at the beginning of the story?

A. To tell the readers he has more than one nephew.

B. To let the readers get an first impression of his nephew.

C. To show the readers how much he loved his nephew.

D. To describe how cruel the boy’s parents were.

49. Which of the following describes the author’s nephew in his childhood?

A. He was clever enough to avoid annoying the adults in his family.

B. He had bent legs and was always cross-eyed.

C. He never shouted or cried on the top of his voice.

D. He was quite tall for his age and good at joking.

50. What do you think the author will Not tell us in the following paragraph?

A. More information about his nephew nowadays.

B. Why his nephew didn’t give a warning before coming.

C. Why his nephew came to his house.

D. What happened to the nephew’s brother.

UNITED NATIONS – Iran's UN Mission sent a letter Thursday requesting that President Mahmoud Ahmadinejad be allowed to speak before the Security Council when it votes on new sanctions(制裁) against Tehran for its refusal to delay uranium（铀）development, the council president said.

South Africa's UN Ambassador Dumisani Kumalo said he would present Iran's request to the 14 other council members Friday. Under the UN Charter and Security Council rules, if a member state has an issue before the council and requests to appear before its members, "this must be considered."

The request from Iran's UN Ambassador Javad Zarif arrived as the council's five

vote-wielding members(常任理事国) and Germany agreed on new sanctions to step up the pressure on Tehran to delay development.

Iran has refused UN’s demands that it stop development, insisting its nuclear program is peaceful and aimed at producing energy. The US and its European allies are concerned its real aim is to produce nuclear weapons.

Earlier Thursday, Ahmadinejad called the Security Council an "illegitimate" body and said any new sanctions forced on his country would only encourage it to be self-sufficient (able to satisfy one’s own needs) and f urther develop nuclear technology.

Acting US Ambassador Alejandro Wolff, reacting to that comment and the possibility of the Iranian president addressing the council, said: "I find it ironic that a president who's

quoted today saying that he tears up Security Council resolutions and has no respect for what the council does, is interested in coming and speaking to the council."

51. Which of the following is the main idea of the passage?

A. Iran’s president hoped to speak before the Security Council for its nuclear program.

B. Iran’s president tried to explain Iran’s nuclear program.

C. Iran has refused UN’s demands about its nuclear program.

D. The UN will vote on new sanctions against Tehran for its uranium development. 52. We can infer from this news that________________.

A. the US Ambassador Wolff doesn’t believe the sincerity of Ahmadinejad

B. there is no doubt that Iran’s unclear program is peaceful

C. Germany is one of the five vote-wielding members

D. the US and its European allies aren’t concerned with Iran’s real aim of nuclear program 53. The underlined word “ironic” in the last paragraph probably means __________. A. strange B. shocking C. hard D. bitterly funny

54. Iran's UN Mission sent a letter to the Security Council because _________________. A. Iran intended to apologize for its uranium development B. Ahmadinejad wanted to address the Security Council C. Iran wanted to become a Security Council member D. the UN had required Iran to do so

55. Where can this passage be probably taken from?

A. A guidebook

B. A fashion magzine

C. The Internet

D. A novel

第Ⅱ卷（共35分）

While it is impossible to live completely free of stress, it is possible to prevent stress as well as reduce its effect when it can’t be avoided. The US Department of Health and Human Services offers the following suggestions for ways to deal with stress. ·1. ________________ When you are nervous, angry or upset, try releasing the pressure through exercise or physical activity. Running, walking, playing tennis, or working in your garden are just some of the activities you might try. Physical exercise will relieve your anxiety and worry and help you to relax. Your body and your mind will work together to ease the stress in your life. ·2. ________________

It helps to talk to someone about your anxieties and worries. Perhaps a friend, family member, teacher or even your leader can help you a better view of what’s troubling you. If you feel your problem is serious, you might seek professional help from a psychologist or a doctor. Knowing when to ask for help is an important step in avoiding serious problems later.

第三部分: 选句填空。根据文章内容在方框内6个选项中选择5个恰当的句子，使文章完整，并将其编号按顺序填入文章后的横线上。（共5小题；每小题1分，共5分）

·3. _________________.

You should make every effort to eat well and get enough rest. If you easily get angry and cannot sleep well enough, or if you’re not eating properly, it will be more li kely that you will fall into stressful situations. If stress repeatedly keep you from sleeping, you should consult a doctor.

·4. __________________

Schedule time for both work and entertainment. Don’t froget, play can be just as important to you as work. You need a break from your daily routine to just relax and have fun. Go window- shopping or work on a hobby. Allow yourself at least a half hour each day to do something you enjoy.

·5. ___________________

Stress can reslut from disorganization and a feeling that “there’s so much to do, and not enough time.” Trying to take care of everything at once can be too much for you and as a result, you may not achieve anything. Instead, make a list of everything you have to do, then do one thing at a time, cheking off each task as it is completed. Set out to do the most important tasks first.

1. ______

2. _______

3. ______

4. _______

5. _______

Some get water from lakes or rivers 3. n________. But some cities have to get water from lakes or rivers far away.

Water is usually 4. b______ to a city from a place higher than the city itself. 5. W______ water is found only in a 6. l_______ place, it must be pumped up to the city. Then it is sent to each home.

Dams are often 7. b_________ in a river to keep the water 8. f_______ flowing away. Then a new lake is formed. People know how much water they need, and 9. a__________ extra water to flow away below the dam. In this way the city stores water for its 10.

u________.

There is usually a building near the dam, 11. w________ much important work is done. For example, the water is tested 12. b_________ it is piped to the city.

We must make the water clean before we 13. d_______ it, sometimes things must be taken away from it, sometimes must be added to it. Machines mix these 14. a_________ things with the water. In this way, it is made drinkable.

In 15. m__________ hotels, you usually find three taps. One is for cold water, another is for hot water and the third is for iced water.

1.____________

2. __________

3. ____________

4. _____________

____________

6. ___________

7. __________

8. ____________

9. _____________ 10.

____________

11. __________ 12. __________ 13. ___________ 14. ____________

15.

____________ 第四部分: 综合填空根据文章内容及所给单词的首字母,写出文中所缺单词，使文章完整、通顺。将完整的单词按序号写在文章下面的横线上。（共15小题；每小题1分，共15分）

第五部分: 书面表达（共15分）

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九年级圆基础知识点--(圆讲义)

一对一授课教案学员姓名：何锦莹年级：9 所授科目：数学一、圆的定义： 1. 描述性定义：在一个平面内，线段OA绕它固定的一个端点O旋转一周，另一个端点A随之旋转所形成的图形叫做圆，其中固定端点O叫做圆心，OA叫做半径． 2 圆的表示方法：通常用符号⊙表示圆，定义中以O为圆心，OA为半径的圆记作“O ⊙”，读作“圆O”． 3 同圆、同心圆、等圆：圆心相同且半径相等的圆叫同圆；圆心相同，半径不相等的两个圆叫做同心圆；能够重合的两个圆叫做等圆. 注意：同圆或等圆的半径相等． 1. 弦：连结圆上任意两点的线段叫做弦． 2. 直径：经过圆心的弦叫做圆的直径，直径等于半径的2倍． 3. 弦心距：从圆心到弦的距离叫做弦心距． 4. 弧：圆上任意两点间的部分叫做圆弧，简称弧．以A B 、为端点的圆弧记作AB，读作弧AB． 5. 等弧：在同圆或等圆中，能够互相重合的弧叫做等弧． 6. 半圆：圆的任意一条直径的两个端点分圆成两条弧，每一条弧都叫做半圆． 7. 优弧、劣弧：大于半圆的弧叫做优弧，小于半圆的弧叫做劣弧． 1. 圆心角：顶点在圆心的角叫做圆心角．将整个圆分为360等份，每一份的弧对应1?的圆心角，我们也称这样的弧为1?的弧．圆心角的度数和它所对的弧的度数相等． 2. 圆周角：顶点在圆上，并且两边都和圆相交的角叫做圆周角． 3. 圆周角定理：一条弧所对的圆周角等于它所对的圆心角的一半．推论1：同弧或等弧所对的圆周角相等；同圆或等圆中，相等的圆周角所对的弧相等．推论2：半圆（或直径）所对的圆周角是直角，90?的圆周角所对的弦是直径．推论3：如果三角形一边上的中线等于这边的一半，那么这个三角形是直角三角形． 4. 圆心角、弧、弦、弦心距之间的关系定理：在同圆或等圆中，相等的圆心角所对的弧相等，所对的弦相等，所对的弦的弦心距相等．推论：在同圆或等圆中，如果两个圆心角、两条弧、两条弦或两条弦的弦心距中有一组量相等，那么它们所对应的其余各组量分别相等．

初中数学竞赛辅导讲义及习题解答_第18讲_圆的基本性质

初中数学竞赛辅导讲义及习题解答学历训练 1．D是半径为5cm的⊙O内一点，且OD＝3cm，则过点D的所有弦中，最小弦AB= ．2．阅读下面材料：对于平面图形A，如果存在一个圆，使图形A上的任意一点到圆心的距离都不大于这个圆的半径，则称图形A被这个圆所覆盖．对于平面图形A，如果存在两个或两个以上的圆，使图形A上的任意一点到其中某个圆的圆心的距离都不大于这个圆的半径，则称图形A被这些圆所覆盖．例如：图甲中的三角形被一个圆所覆盖，图乙中的四边形被两个圆所覆盖．回答下列问题： (1)边长为lcm的正方形被一个半径为r的圆所覆盖，r的最小值是cm； (2)边长为lcm的等边三角形被一个半径为r的圆所覆盖，r的最小值是cm； (3)长为2cm，宽为lcm的矩形被两个半径都为r的圆所覆盖，r的最小值是cm． (2003年南京市中考题) 3．世界上因为有了圆的图案，万物才显得富有生机，以下来自现实生活的图形中都有圆：它们看上去多么美丽与和谐，这正是因为圆具有轴对称和中心对称性． (1)请问以下三个图形中是轴对称图形的有，是中心对称图形的有 (分别用下面三个图的代号a，b，c填空)． (2)请你在下面的两个圆中，按要求分别画出与上面图案不重复的图案(草图) (用尺规画或徒手画均可，但要尽可能准确些，美观些)． a．是轴对称图形但不是中心对称图形． b．既是轴对称图形又是中心对称图形． 4．如图，AB是⊙O的直径，CD是弦，若AB=10cm，CD＝8cm，那么A、B两点到直线CD的距离之和为( ) A．12cm B．10cm C．8cm D．6cm

5．一种花边是由如图的弓形组成的，ACB 的半径为5，弦AB ＝8，则弓形的高CD 为( ) A ．2 B ．25 C ．3 D ．3 16 6．如图，在三个等圆上各自有一条劣弧AB 、CD 、EF ，如果AB+CD=EF ，那么AB+CD 与E 的大小关系是（） A ．AB+CD ＝EF B ．AB+CD=F C ． AB+CD

圆专题讲义

与圆有关的证明及计算 1．已知，如图，直线MN交⊙O于A，B两点，AC是直径，AD平分∠CAM交⊙ O于D，过D作DE⊥MN于E．（1）求证：DE是⊙O的切线；（2）若DE=6cm，AE=3cm，求⊙O的半径． 2．如图，在△ABC，AB=AC，以AB为直径的⊙O分别交AC、BC于点D、E，点 CBF=∠CAB．ACF在的延长线上，且∠ （1）求证：直线BF是⊙O的切线； CBF=，求BC和BF的长．（2）若AB=5，sin∠ 3．如图，四边形ABCD内接于⊙O，BD是⊙O的直径，AE⊥CD，垂足为E，DA

平分∠BDE．（1）求证：AE是⊙O的切线；（2）若∠DBC=30°，DE=1cm，求BD的长．是的中点，过点D作是⊙O的直径，DO4．如图，已知△ABC内接于⊙，AC．ECA 的延长线、F直线BC的垂线，分别交CB、的切线；）求证：EF是⊙O（1 ，求⊙O的半径．EF=8（2）若，EC=6 5．如图，AB是⊙O的直径，弦CD⊥AB与点E，点P在⊙O上，∠1=∠C，（1）求证：CB∥PD； P=，求⊙Osin∠的直径．，（2）若BC=3

6．如图，直线EF交⊙O于A、B两点，AC是⊙O直径，DE是⊙O的切线，且DE⊥EF，垂足为E．（1）求证：AD平分∠CAE；（2）若DE=4cm，AE=2cm，求⊙O的半径． 7．如图，Rt△ABC中，∠ABC=90°，以AB为直径作半圆⊙O交AC与点D，点 E为BC的中点，连接DE．（1）求证：DE是半圆⊙O的切线．（2）若∠BAC=30°，DE=2，求AD的长． 8．如图，在Rt△ABC中，∠ACB=90°，以AC为直径作⊙O交AB于点D点，连接CD．（1）求证：∠A=∠BCD；（2）若M为线段BC上一点，试问当点M在什么位置时，直线DM与⊙O相切？

沪教版-九年级(初三)数学-圆与正多边形讲义-圆的概念及性质复习讲义教案

一、圆的相关概念 1. 圆的定义（1）描述性定义：在一个平面内，线段OA 绕它固定的一个端点O 旋转一周，另一个端点A 随之旋转所形成的图形叫做圆，其中固定端点O 叫做圆心，OA 叫做半径．（2）集合性定义：平面内到定点的距离等于定长的点的集合叫做圆，顶点叫做圆心，定长叫做半径．（3）圆的表示方法：通常用符号⊙表示圆，定义中以O 为圆心，OA 为半径的圆记作”O ⊙“，读作” 圆O “．（4）同圆、同心圆、等圆：圆心相同且半径相等的圆叫同圆；圆心相同，半径不相等的两个圆叫做同心圆；能够重合的两个圆叫做等圆．注意：注意：同圆或等圆的半径相等． 2. 弦和弧（1）弦：连结圆上任意两点的线段叫做弦．（2）直径：经过圆心的弦叫做圆的直径，直径等于半径的2倍．（3）弦心距：从圆心到弦的距离叫做弦心距．（4）弧：圆上任意两点间的部分叫做圆弧，简称弧．以A B 、为端点的圆弧记作AB ，读作弧AB ．（5）等弧：在同圆或等圆中，能够互相重合的弧叫做等弧．（6）半圆：圆的任意一条直径的两个端点分圆成两条弧，每一条弧都叫做半圆．（7）优弧、劣弧：大于半圆的弧叫做优弧，小于半圆的弧叫做劣弧．（8）弓形：由弦及其所对的弧组成的图形叫做弓形． 3. 圆心角和圆周角（1）圆心角：顶点在圆心的角叫做圆心角．将整个圆分为360等份，每一份的弧对应1?的圆心角，我们也称这样的弧为1?的弧．圆心角的度数和它所对的弧的度数相等．（2）圆周角：顶点在圆上，并且两边都和圆相交的角叫做圆周角．二、圆的对称性 1. 旋转对称性中考要求知识点睛圆的概念及性质

第3章圆的基本性质单元复习例题讲义

第3章圆的基本性质单元复习 3.1 圆 3.1.1 圆 ·连接圆上任意两点的线段叫做弦。圆上任意两点之间的部分叫做圆弧，简称弧。 3.1.3 弧、弦、圆心角 AB于D，OE⊥AC于E，，半径为R，，求证∠AOB=∠BOC=∠COA。

3.1.4 圆周角 1、顶点在圆上，且两边都与圆相交的角叫做圆周角。 2、圆周角定理：在同圆或等圆中，同弧或等弧所对的圆周角相等，且都等于这条弧所对的圆心角的一半。推论1：在同圆或等圆中，如果两个圆周角相等，那么它们所对的弧也一定相等。推论2：半圆或直径所对的圆周角是直角，90°的圆周角所对的弦是直径。 3、如果一个多边形的所有顶点都在同一个圆上，那么这个多边形就叫做圆内接多边形，这个圆就叫做多边形的外接圆。求证：①如果三角形一条边上的中线等于这条边的一半，那么这个三角形是直角，∠ACB的平分线交⊙O于D，直径所对的圆周角是直角）（勾股定理）两个圆周角相等，则所对的弧也相等）

3.2 点、直线、圆和圆的位置关系 24.2.1 点和圆的位置关系 1、若⊙O的半径为r，点P到圆心的距离为d，则有：点P在圆外?d>r；点P在圆上?d=r；点P在圆内?d

(完整版)直线与圆专题讲义教师版

一、知识梳理 1．点到直线距离公式：点),(00y x P 到直线:0l ax by c ++= 的距离为：d = 2.已知两条平行线直线1l 和2l 的一般式方程为 1l ：01=++C By Ax ，2l ：02=++C By Ax ，则1l 与2l 的距离为2 2 21B A C C d +-= 3．两条直线的位置关系：直线方程平行的充要条件垂直的充要条件备注 2 22111::b x k y l b x k y l +=+= 21,21b b k k ≠= 121-=?k k 21,l l 有斜率 4. 已知l 1:A 1x+B 1y+C 1=0，l 2:A 2x+B 2y+C 2=0，则l 1 ⊥l 2的充要条件是A 1A 2+B 1B 2=0。 5．圆的方程： ⑴标准方程：①2 2 2 )()(r b y a x =-+- ；②2 22r y x =+ 。 ⑵一般方程：022=++++F Ey Dx y x （)042 2>-+F E D 注：Ax 2+Bxy+Cy 2+Dx+Ey+F=0表示圆?A=C≠0且B=0且D 2+E 2－4AF>0； 6．圆的方程的求法：⑴待定系数法；⑵几何法。 7．点、直线与圆的位置关系：（主要掌握几何法） ⑴点与圆的位置关系：（d 表示点到圆心的距离） ①?=R d 点在圆上；②?R d 点在圆外。 ⑵直线与圆的位置关系：（d 表示圆心到直线的距离） ①?=R d 相切；②?R d 相离。 ⑶圆与圆的位置关系：（d 表示圆心距，r R ,表示两圆半径，且r R >） ①?+>r R d 相离；②?+=r R d 外切；③?+<<-r R d r R 相交； ④?-=r R d 内切；⑤?-<

九年级圆基础知识点,(圆讲义)

一对一授课教案学员姓名：____何锦莹____ 年级：_____9_____ 所授科目：___数学__________ 上课时间：____ 年月日_ ___时分至__ __时_ __分共 ___小时一、圆的定义： 1. 描述性定义：在一个平面内，线段OA绕它固定的一个端点O旋转一周，另一个端点A随之旋转所形成的图形叫做圆，其中固定端点O叫做圆心，OA叫做半径． 2 圆的表示方法：通常用符号⊙表示圆，定义中以O为圆心，OA为半径的圆记作“O ⊙”，读作“圆O”． 3 同圆、同心圆、等圆：圆心相同且半径相等的圆叫同圆；圆心相同，半径不相等的两个圆叫做同心圆；能够重合的两个圆叫做等圆. 注意：同圆或等圆的半径相等． 1. 弦：连结圆上任意两点的线段叫做弦． 2. 直径：经过圆心的弦叫做圆的直径，直径等于半径的2倍． 3. 弦心距：从圆心到弦的距离叫做弦心距． 4. 弧：圆上任意两点间的部分叫做圆弧，简称弧．以A B 、为端点的圆弧记作AB，读作弧AB． 5. 等弧：在同圆或等圆中，能够互相重合的弧叫做等弧． 6. 半圆：圆的任意一条直径的两个端点分圆成两条弧，每一条弧都叫做半圆． 7. 优弧、劣弧：大于半圆的弧叫做优弧，小于半圆的弧叫做劣弧．

8. 弓形：由弦及其所对的弧组成的图形叫做弓形． 1. 圆心角：顶点在圆心的角叫做圆心角．将整个圆分为360等份，每一份的弧对应1?的圆心角，我们也称这样的弧为1?的弧．圆心角的度数和它所对的弧的度数相等． 2. 圆周角：顶点在圆上，并且两边都和圆相交的角叫做圆周角． 3. 圆周角定理：一条弧所对的圆周角等于它所对的圆心角的一半．推论1：同弧或等弧所对的圆周角相等；同圆或等圆中，相等的圆周角所对的弧相等．推论2：半圆（或直径）所对的圆周角是直角，90?的圆周角所对的弦是直径．推论3：如果三角形一边上的中线等于这边的一半，那么这个三角形是直角三角形． 4. 圆心角、弧、弦、弦心距之间的关系定理：在同圆或等圆中，相等的圆心角所对的弧相等，所对的弦相等，所对的弦的弦心距相等．推论：在同圆或等圆中，如果两个圆心角、两条弧、两条弦或两条弦的弦心距中有一组量相等，那么它们所对应的其余各组量分别相等．一、圆的对称性 1. 圆的轴对称性：圆是轴对称图形，对称轴是经过圆心的任意一条直线． 2. 圆的中心对称性：圆是中心对称图形，对称中心是圆心． 3. 圆的旋转对称性：圆是旋转对称图形，无论绕圆心旋转多少角度，都能与其自身重合．二、垂径定理 1. 垂径定理：垂直于弦的直径平分这条弦，并且平分弦所对的两条弧． 2. 推论1：⑴平分弦（不是直径）的直径垂直于弦，并且平分弦所对的两条弧； ⑵弦的垂直平分线经过圆心，并且平分弦所对的两条弧； ⑶平分弦所对的一条弧的直径，垂直平分弦，并且平分弦所对的另一条弧． 3. 推论2：圆的两条平行弦所夹的弧相等．练习题；

初中数学专题讲义-圆(一)

初中数学专题讲义-圆(一) 一、课标下复习指南 1．圆的有关概念圆、圆心、半径、等圆；弦、直径、弦心距、弧、半圆、优弧、劣弧、等弧；三角形的外接圆、三角形的内切圆、三角形的外心、三角形的内心、圆心角、圆周角．2．圆的对称性圆是轴对称图形，任何一条直径所在直线都是它的对称轴，圆有无数条对称轴；圆是以圆心为对称中心的中心对称图形．圆具有旋转不变性． 3．圆的确定不在同一直线上的三个点确定一个圆． 4．垂直于弦的直径垂径定理垂直于弦的直径平分这条弦，并且平分弦所对的两条弧．推论平分弦(不是直径)的直径垂直于弦，并且平分弦所对的两条弧． 5．圆心角、弧、弦之间的关系定理在同圆或等圆中，相等的圆心角所对的弧相等，所对的弦也相等．推论在同圆或等圆中，如果两个圆心角、两条弧、两条弦中有一组量相等，那么它们所对应的其余各组量也相等． 6．圆周角圆周角定理在同圆或等圆中，同弧或等弧所对的圆周角相等，都等于这条弧所对的圆心角的一半．推论1 在同圆或等圆中，相等的圆周角所对的弧也相等．推论2 半圆(或直径)所对的圆周角是直角；90°的圆周角所对的弦是直径．7．点和圆的位置关系设⊙O的半径为r，点P到圆心的距离OP＝d，则有：点P在圆外?d＞r；点P在圆上?d＝r；点P在圆内?d＜r． 8 直线和圆相离相切相交的位置图形公共点的 0 1 2 个数公共点无切点交点名称直线名称无切线割线圆心到直线的距离 d＞r d＝r d＜r d与半径 r的关系

9．切线的判定切线的判定定理经过半径的外端并且垂直于这条半径的直线是圆的切线． (会过圆上一点画圆的切线) 10．切线的性质切线的性质定理圆的切线垂直于过切点的半径． 11．切线长和切线长定理切线长经过圆外一点作圆的切线，这点和切点之间的线段的长，叫做这点到圆的切线长．切线长定理从圆外一点可以引圆的两条切线，它们的切线长相等，这一点和圆心的连线平分两条切线的夹角．二、例题分析例1 已知：如图14－1，在⊙O 中，弦AB 的中点为C ，过点C 的半径为OD ．图14－1 (1)若AB ＝32，OC ＝1，求CD 的长； (2)若半径OD ＝R ，∠AOB ＝120°，求CD 的长．分析圆的半径、弦长的一半、弦心距三条线段组成一个直角三角形，其中一个锐角为弦所对圆心角的一半，可充分利用它们的关系解决有关垂径定理的计算问题．解 ∵半径OD 经过弦AB 的中点C ， ∴半径OD ⊥AB (1)∵AB ＝32，∴AC ＝BC ＝3． ∵OC ＝1，由勾股定理得OA ＝2． ∴CD ＝OD －OC ＝OA －OC ＝1． (2)∵OD ⊥AB ，OA ＝OB ， ∴∠AOD ＝∠BOD ． ∵∠AOB ＝120°，∴∠AOC ＝60°． ,2 160cos cos R OA AOC OA OC = ?=∠?=οΘ .2 1 21R R R OC OD CD =-=-=∴ 说明如图14－1，一般的，若∠AOB ＝2n °，OD ⊥AB 于C ，OA ＝R ，OC ＝h ，则AB ＝2R ·sin n °＝2h ·tan n ° ;222h R -= CD ＝R －h ；的长＝ ?180 πR n 例2 已知：如图14－2，⊙O 中，半径OA ＝4，弦BC 经过半径OA 的中点P ，∠OPC ＝60°，求弦BC 的长．

初三数学讲义

暑假数学（九年级）教学具体授课计划

备注： 1.本授课计划的第一、二、五、六、九、十、十六是对七、八年级重点知识点的回顾与复习。编排次序对应于九年级上册相应知识点，以便更加系统明了地做到知识点之间的融会贯通。

2.教学进程大体按照该计划进行。但在授课过程中，也会根据学生的实际情况，适当调整各知识板块的教学进度，或增补缩减相应的资料。 3.不足之处敬请批评指正。欢迎各位家长、老师提出更合理中肯的建议！第一讲数与式的复习（一）【教学目标】 1. 理解有理数的有关概念，能用数轴上的点表示有理数，会求倒数、相反数、绝对值.理解近似数和有效数字的概念，会将一个数表示成科学记数法的形式。 2. 了解算术平方根、平方根、立方根的概念，会求非负数的算术平方根和实数的立方根。 3. 了解整式的有关概念，理解去括号法则，能熟练进行整式的加减运算.掌握正整数指数幂的运算性质，能在运算中灵活运用各种性质。 4. 了解分式概念，会求分式有意义、无意义和分式值为0时，分式中所含字母的条件，掌握分式的基本性质和分式的变号法则，能熟练地进行分式的通分和约分。【重点难点】重点：概念的理解与区分难点：易混淆，各概念的性质及条件【知识梳理】 1.实数分类：

实数???? ?? ?? ????????????? ???? ????????????????????无限不循环小数负无理数正无理数无理数数有限小数或无限循环小负分数正分数分数负整数零正整数整数有理数 2.数轴：规定了原点、正方向和单位长度的直线。数轴上所有的点与全体实数是一一对应关系，即每个实数都可以用数轴上的一个点表示；反过来，数轴上的每一个点都表示一个实数。 3.实数大小的比较：在数轴上表示的两个数，右边的数总比左边的数大。（1）正数大于零，零大于负数。（2）两正数相比较绝对值大的数大，绝对值小的数小。（3）两负数相比较绝对值大的数反而小，绝对值大小的数反而大。（4）对于任意两个实数a 和b ，①a>b,②a=b,③a

人教版必修二数学圆与方程知专题讲义

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涉及点与圆的位置关系：圆上两点的中垂线一定过圆心涉及直线与圆的位置关系：相切时，利用到圆心与切点的连线垂直直线；相交时，利用到点到直线的距离公式及垂径定理 3.2240D E F +->常可用来求有关参数的范围三、点与圆的位置关系 1.判断方法：点到圆心的距离d 与半径r 的大小关系 d r ?点在圆外 2.涉及最值：（1）圆外一点B ，圆上一动点P ，讨论PB 的最值 min PB BN BC r ==- max PB BM BC r ==+ （2）圆内一点A ，圆上一动点P ，讨论PA 的最值 min PA AN r AC ==- max PA AM r AC ==+ 思考：过此A 点作最短的弦？（此弦垂直AC ） 3.以1122(,),(,)A x y B x y 两点为直径的圆方程为 1212()()()()0x x x x y y y y --+--= 四、直线与圆的位置关系 1.判断方法（d 为圆心到直线的距离）（1）相离?没有公共点?0d r ? （2）相切?只有一个公共点?0d r ?=?= （3）相交?有两个公共点?0d r ?>?< 2.直线与圆相切（1）知识要点 ①基本图形

正多边形与圆一对一辅导讲义

1、了解正多边形的概念，探究正多边形与圆的关系； 2、经历探索正多边形与圆的关系，理解正多边形的性质；第一课时正多边形与圆知识点梳理课前检测 1.圆的半径扩大一倍，则它的相应的圆内接正n边形的边长与半径之比( ) A.扩大了一倍 B.扩大了两倍 C.扩大了四倍 D.没有变化 2.正三角形的高、外接圆半径、边心距之比为( ) A.3∶2∶1 B.4∶3∶2 C.4∶2∶1 D.6∶4∶3 3.正五边形共有__________条对称轴，正六边形共有__________条对称轴. 4.中心角是45°的正多边形的边数是__________. 5.已知△ABC的周长为20,△ABC的内切圆与边AB相切于点D,AD=4,那么BC=__________. 知识梳理正多边形的定义：各角相等，各边相等的多边形叫做正多边形．正多边形的相关概念： ⑴正多边形的中心角；⑵正多边形的中心；⑶正多边形的半径；⑷正多边形的边心距正多边形的性质：

⑴正n 边形的半径和边心距把正n 边形分成2n 个全等的直角三角形； ⑵正多边形都是轴对称图形，正n 边形共有n 条通过正n 边形中心的对称轴； ⑶偶数条边的正多边形既是轴对称图形，也是中心对称图形，其中心就是对称中心．正多边形的有关计算 ⑴正n 边形的每个内角都等于 ()2180n n -??； ⑵正n 边形的每一个外角与中心角相等，等于 360n ? ； ⑶设正n 边形的边长为n a ，半径为R ，边心距为n r ，周长为n P ，面积为n S ，则222180180111 2sin cos 422 n n n n n n n n n n n a R r R R r a P na S n r a r P n n ??===+==??=?，，，，正多边形的画法 1.用量角器等分圆由于在同圆中相等的圆心角所对的弧相等，因此作相等的圆心角可以等分圆. 2.用尺规等分圆对于一些特殊的正ｎ边形，可以用圆规和直尺作图. 第二课时正多边形与圆典型例题题型一、正多边形的概念例1.填写下列表中的空格正多边形边数内角中心角半径边长边心距周长面积 3 23 4 1 6 2 变1.（1）若正n 边形的一个外角是一个内角的 3 2 时，此时该正n 边形有_________条对称轴. 典型例题

数学九年级下沪科版第25.8正多边形和圆讲义教案

正多边形和圆(一）一．内容综述正多边形的有关计算方法、圆及简单组合图形的周长与面积的计算方法，是本单元的重点。实际上，这部分计算问题的解决大都是放在直角三角形（如下图△OAD）中解决的。掌握这些知识，一方面可以为进一步学习打好基础，另一方面这些知识在生产和生活中常常用到，所以要给予足够的重视。在正多边形的有关计算中，如果分别以αn、a n、r n、R n、P n 和S n表示正n(n≥3,n为整数)边形的中心角、边长、边心距、半径、周长和面积，则有： ①αn= ； ②a n=2R n·sin ； ③r n=R n·cos ； ④+ ； ⑤P n=na n； ⑥S n= P n r n； ⑦S n= n sin .（因为一个三角形的面积为：h·OB）注意两点：1、构造直角三角形（弦心距、边长的一半、半径组成的）求线段之间的关系等； 2、准确记忆相关公式。在圆的有关计算中，如果用R表示圆的半径，n表示弧或弧所所对的圆心角的度数，L 表示弧长，则有： ①圆周长：C=2πR。 ②弧长：L= ③圆面积：S=πR2 ④扇形面积：S扇形= = LR ⑤弓形面积可利用扇形面积与三角形面积的和或差来计算需根据不同的情况作出不同的处理：

（1）当弓形所含弧为劣弧时，S弓=S扇-S△ （2）当弓形所含弧为优弧时，S弓=S扇+S△ （3）当弓形所含弧为半圆时，S弓= S圆 ⑥圆柱与圆锥的侧面积可以转化为计算侧面展开图的面积二．例题分析：例1.正六边形两条对边之间的距离是2，则它的边长是（） A、B、C、D、解：如图1，BF=2，过点A作AG⊥BF于G，则FG=1，又∵∠FAG=60°，故选B。说明：正六边形是正多边形中最重要的多边形，要注意正六边形的一些特殊性质。例2.如图2，两个同心圆被两条半径截得的的长为6πcm, 的长为10πcm，若AB=12cm,求图中阴影部分的面积。解：设∠O=α，由弧长公式得6π= , 10π= , ∴OA= , OB= . 又∵AB=OB-OA, ∴12= - , ∴α=60°, ∴OA= =18, OB= =30.

精品九年级数学上册圆的基本性质讲义+同步练习题

圆的基本性质知识点圆的定义几何定义：线段OA，绕O点旋转一周得到的图形，叫做圆。其中，O为圆心，OA为半径。集合定义：到定点等于定长的所有点的集合。其中，定点为圆心，定长为半径。圆的书写格式：圆的对称性（1）圆是轴对称图形，它的对称轴是直径所在的直线。（2）圆是中心对称图形，它的对称中心是圆心。（3）圆是旋转对称图形。与圆有关的线段半径：圆上一点与圆心的连线段。确定一个圆的要素是圆心和半径。弦：连结圆上任意两点的线段叫做弦。直径：经过圆心的弦叫做直径。弦心距：圆心到弦的垂线段的长。弧：圆上任意两点间的部分叫做圆弧，简称弧。劣弧：小于半圆周的圆弧叫做劣弧。表示方法：优弧：大于半圆周的圆弧叫做优弧。表示方法：在同圆或等圆中，能够互相重合的弧叫做等弧。注意：同弧或等弧对应的弦相等。垂径定理：垂直于弦的直径平分这条弦，并且平分弦所对的两条弧。注意:定理中的“垂直于弦的直径”可以是直径，也可以是半径，深圳可以是过圆心的直线或线段；该定理也可以理解为：若一条直线具有两条性质：①过圆心；②垂直于一条弦，则此直线具有另外三条性质：①平分此弦；②平分此弦所对的优弧；③平分此弦所对的劣弧. 推论1：（1）平分弦（不是直径）的直径垂直于弦，并且平分弦所对的两条弧。（2）弦的垂直平分线经过圆心，并且平分弦所对的两条弧。（3）平分弦所对的一条弧的直径，垂直平分弦，并且平分弦所对的另一条弧。推论2：圆的两条平行弦所夹的弧相等。在下列五个条件中:①CD是直径;②CD⊥AB;③AM=BM;④AC=BC;⑤AD=BD.只要具备其中两个条件,就可推出其余三个结论. 注意：(1)在圆中，与已知弦（非直径）相等的弦共有条；共端点且相等的弦共有条。（2）在圆中，与已知弦（非直径）平行的弦共有条；平行且相等的弦共有条。例1.如图：OA、OB为⊙O的半径，C、D分别为OA、OB的中点，求证：AD=BC.

山西省中考数学第15讲正多边形和圆与圆中的计算复习讲义(无答案)

正多边形和圆与圆中的计算模块一正多边形和圆正多边形的定义：__________________________________________________。正多边形的相关概念： ⑴正多边形的中心： _______________________________________________。 ⑵正多边形的半径： _______________________________________________。 ⑶正多边形的中心角： _____________________________________________。 ⑷正多边形的边心距： _____________________________________________。正多边形的性质： ⑴______________________________________________________________； ⑵______________________________________________________________ ______________________________________________________________。

【例1】 ⑴小亮从A点出发前进10m，向右转15°，再前进10m，又向右转15°，……，这样一直走下去，他第一次回到出发点A时，一共走了_________m。 ⑵正二百五十边形的一个内角等于_____，它的中心角等于__________。 ⑶正六边形的边长a，半径R，边心距r的比a∶R∶r＝__________________。【例2】 (浙江杭州中考)如图，有一个圆O和两个正六边形T1、T2。T1的6个顶点都在圆周上，T2的6条边都和圆O相切(我们称T1、T2分别为圆O的内接正六边形和外切正六边形)。 ⑴设T1、T2的边长分别为a、b，圆O的半径为r，求r∶a及r∶b的值； ⑵求正六边形T1、T2的面积比S1∶S2的值。模块二圆中的计算设⊙O的半径为R，n°圆心角所对弧长为l 1．弧长公式：____________________。 2．扇形面积公式：______________________。 3．圆柱体表面积公式：______________________。

2021年初中数学竞赛辅导讲义及习题解答第 8讲圆的基本性质

感谢您使用本资源，本资源是由订阅号”初中英语资源库“制作并分享给广大用户，本资源制作于2020年底，是集实用性、可编辑性为一体。本资源为成套文件，包含本年级本课的相关资源。有教案、教学设计、学案、录音、微课等教师最需要的资源。我们投入大量的人力、物力，聘请精英团队，从衡水中学、毛毯厂中学、昌乐中学等名校集合了一大批优秀的师资，精研中、高考，创新教学过程，将同学们喜闻乐见的内容整体教给学生。本资源适用于教师下载后作为教学的辅助工具使用、适合于学生家长下载后打印出来作为同步练习使用、也适用于同学们自己将所学知识进行整合，整体把握进度和难度，是一个非常好的资源。如果需要更多成套资料，请微信搜索订阅号“初中英语资源库”，在页面下方找到“资源库”，就能得到您需要的每一份资源（包括小初高12000份主题班会课课件免费赠送！）第十八讲圆的基本性质到定点(圆心)等于定长(半径)的点的集合叫圆，圆常被人们看成是最完美的事物，圆的图形在人类进程中打下深深的烙印．圆的基本性质有：一是与圆相关的基本概念与关系，如弦、弧、弦心距、圆心角、圆周角等；二是圆的对称性，圆既是一个轴对称图形，又是一中心对称图形．用圆的基本性质解题应注意： 1．熟练运用垂径定理及推论进行计算和证明； 2．了解弧的特性及中介作用； 3．善于促成同圆或等圆中不同名称等量关系的转化．熟悉如下基本图形、基本结论：【例题求解】【例1】在半径为1的⊙O中，弦AB、AC的长分别为3和2，则∠BAC度数为．作出辅助线，解直角三角形，注意AB与AC有不同的位置关系．注：由圆的对称性可引出许多重要定理，垂径定理是其中比较重要的一个，它沟通了线段、角与圆弧的关系，应用的一般方法是构造直角三角形，常与勾股定理和解直角三角形知识结合起来．圆是一个对称图形，注意圆的对称性，可提高解与圆相关问题周密性．【例2】如图，用3个边长为1的正方形组成一个对称图形，则能将其完全覆盖的圆的最小半径为( )

与圆有关的位置关系(讲义)

与圆有关的位置关系（讲义）?知识点睛 1.点与圆的位置关系 d表示__________的距离，r表示___________． ①点在圆外?_____________； ②点在圆上?_____________； ③点在圆内?_____________．三点定圆定理：_________________________________．注：三角形的三个顶点确定一个圆，这个圆叫做三角形的外接圆，外接圆的圆心是三角形三条边垂直平分线的交点，叫做三角形的外心． 2.直线与圆的位置关系 d表示__________________的距离，r表示__________． ①直线与圆相交?____________； ②直线与圆相切?____________； ③直线与圆相离?____________．切线的判定定理：__________________________________ __________________________________________________；切线的性质定理：__________________________________．*切线长定理：______________________________________ __________________________________________________．注：与三角形各边都相切的圆叫做三角形的内切圆，内切圆的圆心是三角形三条角平分线的交点，叫做三角形的内心．*3. 圆与圆的位置关系 d表示__________的距离，R表示________，r表示 _________． ①圆与圆外离?_________________； ②圆与圆外切?_________________； ③圆与圆内切?_________________； ④圆与圆内含?_________________； ⑤圆与圆相交?_________________． 4.圆内接正多边形 _______________________________叫做圆内接正多边形，这个圆叫做该正多边形的_________. 正多边形的中心：___________________________________；正多边形的半径：___________________________________； A

精品 2014年九年级数学圆的基本性质圆周角圆心角讲义+同步练习题

九年级数学圆周角圆心角知识点：圆心角：弧度：圆周角：圆心角与圆周角的关系: 同圆或等圆中，同弧或等弧所对的圆周角等于它所对的圆心角的一半．圆周角定理：直径所对的圆周角是直角，反过来，90°的圆周角所对的弦是直径。例1.如图，已知P 是O 外任意一点，过点P 作直线PAB ，PCD ，分别交O 于点A ，C ，D ．求证：1 2 P ∠= （BD 的度数AC -的度数）．例2.如图①，点A 、B 、C 在⊙O 上，连结OC 、OB ： ⑴ 求证：∠A=∠B+∠C ；⑵ 若点A 在如图②的位置，以上结论仍成立吗？说明理由。例3.如图,⊙O 的直径AB=8cm,∠CBD=300 ,求弦DC 的长. 30? D C B A O

例4.如图所示，已知AB 为⊙O 的直径，CD 是弦，且AB CD 于点E ．连接AC 、OC 、BC ．（1）求证：∠ACO=∠BCD ；（2）若EB=8cm ，CD=24cm ，求⊙O 的直径．例5.如图,在⊙O 中,AB 是直径,CD 是弦,AB ⊥CD. (1)P 是CAD 上一点(不与C 、D 重合),试判断∠CPD 与∠COB 的大小关系, 并说明理由. (2)点P / 在劣弧CD 上(不与C 、D 重合时),∠CP / D 与∠COB 有什么数量关系?请证明你的结论. D C B P A O 例6.如图,A 、B 、C 、D 四点都在⊙O 上,AD 是⊙O 的直径,且AD=6cm,若∠ABC=∠CAD,求弦AC 的长. D C B A O 例7.如图所示，在△ABC 中，∠BAC 与∠ABC 的平分线AE 、BE 相交于点E ，延长AE 交△ABC 的外接圆于 D 点，连接BD 、CD 、C E ，且∠BDA=600 . (1)求证△BDE 是等边三角形；(2) 若∠BDC=1200 ，猜想BDCE 是怎样的四边形，并证明你的猜想。

正多边形与圆弧长扇形计算讲义教案

正多边形的定义：各角相等，各边相等的多边形叫做正多边形．正多边形的相关概念： ⑴正多边形的中心角；⑵正多边形的中心；⑶正多边形的半径；⑷正多边形的边心距正多边形的性质： ⑴正n 边形的半径和边心距把正n 边形分成2n 个全等的直角三角形； ⑵正多边形都是轴对称图形，正n 边形共有n 条通过正n 边形中心的对称轴； ⑶偶数条边的正多边形既是轴对称图形，也是中心对称图形，其中心就是对称中心．正多边形的有关计算 ⑴正n 边形的每个内角都等于 ()2180n n -??； ⑵正n 边形的每一个外角与中心角相等，等于 360n ? ； ⑶设正n 边形的边长为n a ，半径为R ，边心距为n r ，周长为n P ，面积为n S ，则222180180111 2sin cos 422 n n n n n n n n n n n a R r R R r a P na S n r a r P n n ??===+==??=?，，，，正多边形的画法 1.用量角器等分圆由于在同圆中相等的圆心角所对的弧相等，因此作相等的圆心角可以等分圆. 2.用尺规等分圆对于一些特殊的正ｎ边形，可以用圆规和直尺作图. 精典例题：

若它们的面积都相等，那么_____________的周长最大．例题3：在半径为1的圆中有一内接多边形，若它的各边长均大于1且小于2，则这个多边形的边数必为___________．例题4：下面给出六个命题： ①各角相等的圆内接多边形是正多边形；②各边相等的圆内接多边形是正多边形； ③正多边形是中心对称图形；④各角均为120 的六边形是正六边形； ⑤边数相同的正n边形的面积之比等于它们边长的平方比；⑥各边相等的圆外切多边形是正多边形其中，错误的命题是_____________．例题5：（1）正n边形内接于半径为R的圆，这个n边形的面积为2 3R，则n等于____________．（2）正八边形每一个外角是多数等于_______．N边形每一个内角等于________．例题6：(09浙江台州)O ⊙的内接多边形周长为3，O ⊙的外切多边形周长为3.4，则下列各数中与此圆的周长最接近的是( ) A．6 B．8 C．10 D．17 例题7：已知圆内接正六边形面积为33，求该圆外切正方形边长．例题8：已知圆内接正方形的面积为2，求该圆的外切正三角形的外接圆的外切正六边形的面积．强化训练 1、正六边形的两条平行边之间的距离为1，则它的边长为。 2、（2012?天津）若一个正六边形的周长为24，则该六边形的面积为。 3、（2012?巴中）已知一个圆的半径为5cm，则它的内接六边形的边长为。 4、（2012?无锡）如图的平面直角坐标系中有一个正六边形ABCDEF，其中C、D的坐标分别为（1，0）和（2，0）．若在无滑动的情况下，将这个六边形沿着x轴向右滚动，则在滚动过程中，这个六边形的顶点A、B、C、D、E、F中，会过点（45，2）的是点． 5、（2011?绵阳）如图，将正六边形ABCDEF放在直角坐标系中，中心与坐标原点重合，若A点的坐标为（-1，0），则点C的坐标为。 6、（2007?芜湖）如图，PQ=3，以PQ为直径的圆与一个以5为半径的圆相切于点P，正方形ABCD的顶点A、B在大圆上，小圆在正方形的外部且与CD切于点Q．则AB= 。． 7、（2007?天水）如图，已知在⊙O中，直径MN=10，正方形ABCD的四个顶点分别在⊙O及半径OM、OP上，并且∠POM=45°，则AB的长为

专题五 圆中综合应用讲义