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初二下《多边形习题精选》

初二下《多边形习题精选》
初二下《多边形习题精选》

第二十章:《四边形》习题精选

1.如图19-1-28,在ABCD 中,E ,F 为BD 上的点,BF=DE ,那么四边形AECF

是什么图形?试用两种方法证明。

2.已知:在△ABC 中,AB=AC ,EF 是△ABC 的中位线,分别交AB ,AC 于E ,F ,

延长AB 到D ,使BD=AB ,连接CD 。求证:12CE CD

3.如图19-1-29,ABCD 中,对角线AC 、BD 相交于点O ,过点O 作两条直线分

别与AB ,BC ,CD ,AD 交于G ,F ,H ,E 四点。求证:四边形EGFH 是平行四边形。

4.如图19-1-30,分别以△ABC 的三边为边长,在BC 的同侧作等边三角形ABD ,等边三角形BCE ,等边三角形ACF ,连接DE ,EF 。求证:四边形ADEF 是平行四边形。

5.如图19-1-31,在ABCD 中,AE ⊥BD ,CF ⊥BD ,垂足分别为点E ,F ,点G ,H 分别为AD ,BC 的中点,试证明EF 和GH 互相平分。

6.如图19-1-32,△ABC是边长为4cm的边三角形,P是△ABC内的任意一点,过点P作EF∥AB分别交AC,BC于点E,F,作GH∥BC分别交AB,AC于点G,H,作MN∥AC分别交AB,BC于点M,N,试猜想:EF+GH+MN的值是多少?其值是否随P

位置的改变而变化?并说明你的理由。

7.(2005年苏州市)如图19-1-33,在ABCD中,下列各式不一定正确的是()。

A.∠1+∠2=180°

B.∠2+∠3=180°

C.∠3+∠4=180°

D.∠2+∠4=180°

8,已知:如图,在□ABCD 中,E、F分别为边AB、CD的中点,BD是对角线,AG∥DB交CB的延长线于G.

(1) 求证:△ADE≌△CBF;

(2) 若四边形BEDF是菱形,则四边形AGBD是

什么特殊四边形?并证明你的结论.

9.如图,在中,AE平分∠BAD交DC于点E,AD=5cm,AB=8cm,求EC的长.

10ABCD中,AC⊥AB,AB=6,BC=10,求:(1)AB与CD的距离;(2)

AD与BC的距离.

11.如图,已知四边形ABCD是平行四边形,∠BCD的平分线CF交AB于点F,∠ADC 的平分线DG交边AB于点G.

(1)求证:AF=GB;

(2)请你在已知条件的基础上再添加一个条件,使得△EFG为等腰直角三角形,并说明理由.

12.有两张全等的三角形纸片,三角形纸片的三条边长分别为2cm,3cm,4cm.?你能用这两张三角形纸片拼出几种形状不同的平行四边形?

(1)请画图说明各种不同拼法,并说明理由:

(2)计算所拼的各个平行四边形的周长.

13ABCD的面积四等分,并简要说明理由.

14ABCD中,AD⊥DB,AC与BD相交于点O,OD=1,∠CAD=30°,求

AC和DC的长.

15ABCD 中,AC ,BD 交于点O ,EF 过点O ,分别交CB ,AD ?的延长线于点E ,F ,求证:AE=CF .

16、如图,E 是菱形ABCD 边AD 的中点,EF ⊥AC 于点H ,交CB 延长线于点F ,交AB 于点G ,求证:AB 与EF 互相平分。

17、如图,菱形ABCD 中,E 、F 分别为BC 、CD 上的点,且∠B =∠EAF =60°,若∠BAE =20°,求∠CEF 的度数。

18、如图,AD 是△ABC 的角平分线,AD 的垂直平分线交AB 于点E ,交AC 于点F ,求证:四边形AEDF 是菱形。

19、如图,在ABCD 中,BC =2AB ,AE =AB =BF ,且点E 、F 在直线AB 上, 求证:CE ⊥DF 。

20、如图,在Rt △ABC 中,∠ACB =90°,∠BAC =60°,DE 垂直平分BC ,垂足为D ,交AB 于点E ,又点F 在DE 的延长线上,且AF =CE ,求证:四边形ACEF 是菱形。

21、菱形两邻角的比为1∶2,边长为2,求该菱形的面积。

22、已知4

4

4

4

4a b c d abcd +++=,判定以a 、b 、c 、d 为边的四边形的形状。

23、如图,过ABCD 的对角线交点O ,作互相垂直的两条直线EG 、FH 与ABCD 各边分别相交于点E 、F 、G 、H ,求证:四边形EFGH 是菱形。

H G

F

E

D

C B

A F

E

D

B A

F E

D

C

B

A

N

M

F

E D C

B

A F

E D

C

B

A

24、如图,在菱形ABCD 中,E 是AB 中点,且DE ⊥AB ,AB =4,求:(1)∠ABC 的度数;(2)菱形ABCD 的面积。

25、如图,菱形ABCD 的边长为4cm ,且∠ABC =120°,E 是BC 的中点,在BD 上求点P ,使PC +PE 取最小值,并求这个最小值。

26、如图,在ABCD 中,AB =2BC ,BE ⊥AD 于E ,F 为CD 中点,设∠DEF =α,∠EFC =β,求证:β=3α

27、如图,有一矩形纸片ABCD ,AB =6,BC =8,将纸片沿EF 折叠,使点B 与D 重合,求折痕EF 的长。

28、如图,四边形ABCD 中,∠ADC =90°,AC =CB ,E 、F 分别是AC 、AB 的中点,且∠DEA =∠ACB =45°,BG ⊥AE 于G ,求证:(1)四边形AFGD 是菱形;(2)若AC =BC =10,求菱形的面积。

29,如图,矩形ABCD 的对角线相交于点O ,OF ⊥BC ,CE ⊥BD ,OE :BE=1:3,OF=4,求∠ADB 的度数和BD 的长。

O H

G

F

E D

C B A

E

D

C B A

P

E D

C

B

A

F E D

C

B

A

F

E

D

C

B

A

G

F

E

D C

B

A O

F

E

D

C

B

A

30,如图所示,矩形ABCD 中,M 是BC 的中点,且MA ⊥MD ,若矩形的周长为36cm ,求此矩形的面积。

31,折叠矩形纸片ABCD ,先折出折痕BD ,再折叠使AD 边与对角线BD 重合,得折痕DG ,如图,若AB=2,BC=1,求AG 。

32,已知:如图,平行四边形ABCD 的四个内角的平分线分别相交于点E ,F ,G ,H ,求证:四边形EFGH 是矩形。

33,如图,在矩形ABCD 中,E 是AD 上一点,F 是AB 上一点,EF CE =,且

,2EF CE DE cm ⊥=,矩形ABCD 的周长为16cm ,求AE 与CF 的长.

34,如图,在矩形ABCD 中,对角线AC 、BD 相交于点O ,(1),画出△AOB 平移后的三角形,其平移的方向为射线AD 的方向,平移的距离为线段AD 的长。(2)观察平移后的图形,除了矩形ABCD 外还有哪一种特殊的平行四边形?并给出证明。

G

E

D

C

B

A

35,如图所示,已知菱形ABCD 中,E 、F 分别在BC 和CD 上,且∠B=∠EAF=60°,∠BAE=15°,求∠CEF 的度数。

36,已知:如图,在菱形ABCD 中,E 、F 分别是BC 、CD 上的点,且CE=CF 。过点C 作CG ∥EA 交AF 于H ,交AD 于G ,若∠BAE=25°,∠BCD=130°,求∠AHC 的度数。

37,如图所示,已知菱形ABCD 中E 在BC 上,且AB=AE ,∠BAE=2

1

∠EAD ,AE 交BD 于M ,试说明BE=AM 。

38,已知:如图,在矩形ABCD 中,E 、F 分别是边BC 、AB 上的点,且EF =ED ,EF ⊥ED .求证:AE 平分∠BAD .

39,已知:如图,△ABC 中,∠BAC 的平分线交BC 于点D ,E 是AB 上一点,且AE=AC ,EF ∥BC 交AD 于点F ,求证:四边形CDEF 是菱形。

H

G F E

D

C B A

40, 如图,平行四边形ABCD 的对角线AC 的垂直平分线与AD 、BC 、AC 分别交于点E 、F 、O ,求证:四边形AFCE 是菱形。

41,已知:如图,C 是线段BD 上一点,△ABC 和△ECD 都是等边三角形,R 、F 、G 、H 分别是四边形ABDE 各边的中点,求证:四边形RFGH 是菱形。

42,如图,已知在△ABC 中,AB=AC ,∠B ,∠C 的平分线BD 、CE 相交于点M ,DF ∥CE ,EG ∥BD ,DF 与EG 交于N ,求证:四边形MDNE 是菱形。

43,已知:如图所示,ABCD 为菱形,通过它的对角线的交点O 作AB 、BC 的垂线,与AB 、BC ,CD ,DA 分别相交于点E 、F 、G 、H ,求证:四边形EFGH 为矩形。

R H

G

F

E D

C

B

A

44,如图,在四边形ABCD中,点E、F分别是AD、BC的中点,G、H分别是BD、

AC的中点,AB、CD满足什么条件时,四边形EGFH是菱形?请证明你的结论。

45如图,菱形ABCD的边长为2,BD=2,E、F分别是边AD,CD上的两个动点,且满足AE+CF=2.

(1)求证:△BDE≌△BCF;

(2)判断△BEF的形状,并说明理由;

(3)设△BEF的面积为S,求S的取值范围.

46、如图,在平行四边形ABCD中,E是CD的中点,△ABE是等边三角形,求证:四边形ABCD是矩形。

47、已知如图,菱形ABCD中,E是AB的中点,且DE⊥AB,AE=2。

求:(1)∠ABC的度数;(2)对角线AC、BD的长;(3)菱形ABCD的面积。

F

E

D

C B

A

48、如图①,四边形ABCD 是正方形, 点G 是BC 上任意一点,DE ⊥AG 于点E ,BF ⊥AG 于点F .

(1) 求证:DE -BF = EF .

(2) 当点G 为BC 边中点时, 试探究线段EF 与GF 之间的数量关系, 并说明理由. (3) 若点G 为CB 延长线上一点,其余条件不变.请你在图②中画出图形,写出此时DE 、BF 、EF 之间的数量关系(不需要证明).

49、已知:如图,在□ABCD 中,O 为边AB 的中点,且∠AOD=∠BOC .求证:□ABCD 是矩形.

50、已知菱形ABCD 中,AC 与BD 相交O 点,若∠BDC=030,菱形的周长为20厘米,求菱形的面积.

51.如图,在□ABCD 中,AE ⊥BC 于E ,AF ⊥CD 于F ,若AE=4,AF=6,□ABCD 的周长为40。

⑴求BC 的长; ⑵求□ABCD 的面积.

A

B C

D

O B

A C

D

O

52.已知,如图,矩形ABCD 的对角线AC ,BD 相交于点O ,E ,F 分别是OA ,OB 的中点. (1)求证:△ADE ≌△BCF ;(2)若AD=4cm ,AB=8cm ,求OF 的长.

53.如图,矩形ABCD 中,AB =4cm ,BC =8cm ,动点M 从点D 出发,按折线DCBAD 方向以2cm/s 的速度运动,动点N 从点D 出发,按折线DABCD 方向以1cm/s 的速度运动. (1)若动点M 、N 同时出发,经过几秒钟两点相遇?

(2)若点E 在线段BC 上,且BE =3cm ,若动点M 、N 同时出发,相遇时停止运动,经过几秒

钟,点A 、E 、M 、N 组成平行四边形?

54.如图,在正方形ABCD 中,E 是AB 上一点,F 是AD 延长线上一点,且DF =BE . ⑴求证:CE =CF ;

⑵在图1中,若G 在AD 上,且∠GCE =45°,则GE =BE +GD 成立吗?为什么?

55. 已知:如图,在正方形ABCD ,E 是BC 边上一点,F 是CD 的中点,且AE = DC + CE . 求证:AF 平分∠DAE . 法一:

法二:

56.证明:1)对于任意的对角线垂直的凸四边形,连接各边中点,得到的是矩形; 2)对于任意的对角线相等的凸四边形,连接各边中点,得到的是菱形。

B C

D

F E

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a fever?No,I don't./What should I do? You should take your temperature./ Should I put some medicine on it? Yes,you should.等。让学生知道怎样表达身体的不适及正确地处理生活中的一些事情。在学习过程中,学生在交流中,能促进师生之间的感情。Section A 主要学习怎样表达身体的不适并给出合理性的建议。应掌握句型:What's the matter? I have a stomachache.What should I do?等。短文“Bus Driver and Passengers Save an Old Man”介绍了一位公共汽车司机及乘客救一位老人的故事,增加了学生的阅读量。Section B 安排了听、说、读、写的任务,教师在教学中应合理利用课本上的知识进行教学。 第一课时Section A(1a-2d) Teaching Key Points【教学重点】 The vocabulary: matter,throat,foot,stomach,toothache,headache,have a stomachache,have a cold,lie down,take one's temperature,have a fever,go to a doctor Target language: 1.What's the matter? I have a stomachache. 2.What should I do? Should I take my temperature? 3.I think you should lie down and rest. Teaching Difficult Points【教学难点】 Use the target language above to talk about health problems and give advice. Teaching Aids【教学工具】 an English book,a tape recorder and CAI Teaching Steps【教学过程】 ★Step 1Preview and perception【预习感知】 Ask the students to read the vocabulary and target language. 根据句意及汉语或首字母提示完成句子。 1.—What's the matter with her? —She has a very sore t______ now. 2.He ate too much,so he had a s______. 3.If you feel tired,you should l______ down and rest. 4.If you ______(咳嗽),drink some hot tea with honey. 5.He wants to see a dentist,because he has a ______(牙疼). ★Step 2Consociation and exploration【合作探究】 Let the students read the book by themselves in order to find out the answers.They can discuss the questions in groups or ask the teacher for help.When they finish the questions,ask some students to check the answers. ★Step 3Leading in【情景导入】 Ask a student to act something is wrong with his/ her head… And T:What's the matter? Help the students to answer:I have a… Have the students repeat. ★Step 4Pre-task【准备任务】 Page 1,1a &1b

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