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【324296】2024八年级数学下册 专题突破 第07讲 三角形的中位线专题复习(含解析)(新版)浙

时间:2025-01-15 21:54:09 作者: 字数:22475字
简介:


 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 7讲三角形的中位线专题探究

类型一三角形中位线定理

知识点睛:

三角形中位线定理的应用

1)证明平行问题;

2)证明一边是另一边的2倍或

3)解决"中点问题".

注意∶在处理这些问题时,要求出现三角形及其中位线:

有中点连线而无三角形,要作辅助线产生三角形;

有三角形而无中位线,要作中点的连线或过中点作平行线.

类题训练

1.(罗湖区校级期末)如图,△ABC的面积是16,点DEFG分别是BCADBECE的中点,则△AFG的面积是(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A6 B7 C8 D9

【分析】根据中线的性质,可得:△AEF的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×△ABE的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×△ABD的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×△ABC的面积=2,△AEG的面积=2,根据三角形中位线的性质可得△EFG的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×△BCE的面积=2,进而得到△AFG的面积.

【解答】解:∵点DBC的中点,

AD是△ABC的中线,

∴△ABD的面积=△ADC的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×△ABC的面积,

同理得:△AEF的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×△ABE的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×△ABD的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×△ABC的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×162

AEG的面积=2

BCE的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×△ABC的面积=8

又∵FG是△BCE的中位线,

∴△EFG的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×△BCE的面积= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×82

∴△AFG的面积是2×36

故选:A

2.(寿光市期末)如图,DE是△ABC的中位线,∠ABC的角平分线交DE于点FAB8BC12,则EF的长为(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A1 B1.5 C2 D2.5

【分析】延长AFBCH,由三角形中位线定理得到DEBCDE <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BC6AFFH,再证△BFA≌△BFHAAS),得BHAB8,然后由三角形中位线定理得DF4,求解即可.

【解答】解:连接AF并延长交BCH,如图所示:

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> DE分别为边ABAC的中点,

DEBCDE <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BC6AFFH

在△BFA和△BFH中,

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∴△BFA≌△BFHAAS),

BHAB8

ADDBAFFH

DF是△ABH的中位线,

DF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BH4

EFDEDF2

故选:C

3.(海阳市期末)如图,△ABC中,点DE在边BC上,∠ABC的平分线垂直AE,垂足为点N,∠ACB的平分线垂直AD,垂足为点M,连接MN.若BC7MN <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ,则△ABC的周长为(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A17 B18 C19 D20

【分析】利用ASA定理证明△BNA≌△BNE,根据全等三角形的性质得到BEBAANNE,同理得到CDCAAMMD,根据三角形中位线定理求出DE,根据三角形的周长公式计算,得到答案.

【解答】解:在△BNA和△BNE中,

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∴△BNA≌△BNEASA),

BEBAANNE

同理,CDCAAMMD

AMMDANNEMN <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

DE2MN3

BE+CDBCDE

AB+ACBC+DE10

∴△ABC的周长=AB+AC+BC10+717

故选:A

4.(江干区期末)如图,△ABC中,DBC边的中点,AE平分∠BACBEAEE,已知AB10AC18,则DE的长为(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A4 B5 C6 D7

【分析】延长BEACF,证明△AEF≌△AEB,根据全等三角形的性质得到AFAB10BEEF,根据三角形中位线定理计算即可.

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 解答】解:延长BEACF

BEAE

∴∠AEB=∠AEF90°

在△AEF和△AEB中,

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∴△AEF≌△AEBASA

AFAB10BEEF

CFACAF8

BEEFBDDC

DE <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> CF4

故选:A

5.(吴兴区二模)如图,在△ABC中,点DEF分别是各边的中点,若△ABC的面积为4cm2,则△DEF的面积是(  )cm2

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A0.5 B1 C2 D4

【分析】根据三角形中位线定理得到EF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ABED <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ACDF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BC,进而证明△EFD∽△ABC,根据相似三角形的面积比等于相似比的平方计算即可.

【解答】解:∵点DEF分别是各边的中点,

EF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ABED <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ACDF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BC

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∴△EFD∽△ABC,且相似比为 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> =( <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 2 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∵△ABC的面积为4cm2

∴△DEF的面积是1cm2

故选:B

6.(广饶县期末)如图,AD是△ABC的中线,EAD的中点,FBE延长线与AC的交点,若AC4,则AF=(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> B <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> C1 D <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

【分析】取BF的中点H,连接DH,根据三角形中位线定理得到DH <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> FCDHAC,证明△AEF≌△DEH,根据全等三角形的性质得到AFDH,计算即可.

【解答】解:取BF的中点H,连接DH

BDDCBHHF

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> DH <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> FCDHAC

∴∠HDE=∠FAE

在△AEF和△DEH中,

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∴△AEF≌△DEHASA),

AFDH

AF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> FC

AC4

AF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

故选:B

7.(龙口市期末)如图,△ABC的周长为a,以它的各边的中点为顶点作△A1B1C1,再以△AB1C1各边的中点为顶点作△A2B2C2,…如此下去,则△AnBnn的周长为(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a B <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a C <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a D <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a

【分析】根据三角形中位线定理得到△A1B1C1的周长= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a,△A2B2C2的周长= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a,总结规律,根据规律解答即可.

【解答】解:∵点A1B1C1分别为BCACAB的中点,

B1C1 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BCA1C1 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ACA1B1 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AB

∴△A1B1C1的周长= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a

同理,△A2B2C2的周长= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a

……

则△AnBnn的周长= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a

故选:A

8.(东莞市校级期末)如图,已知△ABCABACAD是∠BAC的平分线,AE是∠BAC的外角平分线,EDABAC于点G,下列结论:①ADBC;②AEBC;③AEAG;④∠DAE90°.其中正确结论的个数是(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A1 B2 C3 D4

【分析】连接EC,根据等腰三角形的性质得出ADBC,即可判断①;求出∠FAE=∠B,再根据平行线的性质得出AEBC,即可判断②;求出四边形ABDE是平行四边形,根据平行四边形的性质得出AEBD,求出AECD,根据矩形的判定推出四边形ADCE是矩形,根据矩形的性质得出ACDEAGCGDGEG,求出DGAGCGEG,根据勾股定理判断④即可;根据AEBD <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BCAG <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AC判断③即可.

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 解答】解:连接EC

ABACAD是∠BAC的平分线,

ADBC,故①正确;

ABAC

∴∠B=∠ACB

AE平分∠FAC

∴∠FAC2∠FAE

∵∠FAC=∠B+∠ACB

∴∠FAE=∠B

AEBC,故②正确;

AEBCDEAB

四边形ABDE是平行四边形,

AEBD

ABACADBC

CDBD

AECD

AEBC,∠ADC90°

四边形ADCE是矩形,

∴∠DAE90°,故④正确;

AEBD <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BCAG <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AC

AGAE错误(已知没有条件ACBC),故③错误;

即正确的个数是3个,

故选:C

类型二三角形中位线在四边形中的应用

知识点睛:

四边形中中位线的构造

  1. 四边形边上有中点时,取其对角线中点构造三角形中位线;

  2. 四边形对角线上有中点时,取边的中点构造三角形中位线.

此类中位线的构造常出现在等对边四边形或等对角线四边形题目中,用于判断线段关系或由线段引发的角度关系。

注意∶构造出的中位线往往是相等的,且正好是等对边或等对角线的一半.

类题训练

1 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> .(孟津县期末)如图所示,已知四边形ABCDRP分别是DCBC上的点,点EF分别是APRP的中点,当点P在边BC上从点B向点C移动,且点R从点D向点C移动时,那么下列结论成立的是(  )

A.线段EF的长逐渐增大

B.线段EF的长逐渐减少

C.线段EF的长不变

D.△ABP和△CRP的面积和不变

【分析】连接AR,根据三角形的中位线定理可得EF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AR,根据AR的变化情况即可判断.

【解答】解:连接AR

EF分别是APRP的中点,

EF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AR

当点PBC上从点C向点B移动,点R从点D向点C移动时,AR的长度逐渐增大,

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 线段EF的长逐渐增大.

SABP+SCRP <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BCAB+CR).

CR随着点R的运动而减小,

∴△ABP和△CRP的面积和逐渐减小.

观察选项,只有选项A符合题意.

故选:A

2.(新城区校级期末)如图,四边形ABCD中,ADBC,点P是对角线BD的中点,EF分别是ABCD的中点,若∠EPF130°,则∠PEF的度数为(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A25° B30° C35° D50°

【分析】根据三角形中位线定理得到PF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BCPE <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AD,进而证明PFPE,根据等腰三角形的性质、三角形内角和定理计算,得到答案.

【解答】解:∵PF分别是BDCD的中点,

PF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BC

同理可得:PE <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AD

ADBC

PFPE

∵∠EPF130°

∴∠PEF=∠PFE <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ×180°﹣130°)=25°

故选:A

3.(南阳模拟)如图,四边形ABCD中,ADBCAD2BC5,点EF分别是对角线ACBD的中点,则EF的长为(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A1 B1.5 C2.5 D3.5

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 分析】延长FECD于点G,由点EF分别是对角线ACBD的中点,从而得FG是△BCD的中位线,则有FG2.5,再由ADBC,则有FGADEG是△ACD的中位线,则有EG1,从而可求EF的长.

【解答】解:∵取DC中点G,连结FGEG,如图所示:

EF分别是对角线ACBD的中点,

FGBCEGAD

ADBC

EGBCFGEG

EFG三点共线,

FG是△BCD的中位线,

FG <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BC2.5

ADBC

EGAD

EG是△ACD的中位线,

EG <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AD1

EFFGEG1.5

故选:B

4.(龙岗区校级期末)如图,四边形ABCD中,EF分别是边ABCD的中点,则ADBCEF的关系是(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

AAD+BC2EF BAD+BC≥2EF CAD+BC2EF DAD+BC≤2EF

【分析】取AC的中点G,连接EFEGGF,根据三角形中位线定理求出EG <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BCGF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AD,再利用三角形三边关系:两边之和大于第三边,即可得出ADBCEF的关系.

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 解答】解:如图,取AC的中点G,连接EFEGGF

EF分别是边ABCD的中点,

EGGF分别是△ABC和△ACD的中位线,

EG <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BCGF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AD

在△EGF中,由三角形三边关系得EG+GFEF,即 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BC+ <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ADEF

AD+BC2EF

ADBC时,点EFG在同一条直线上,

AD+BC2EF

所以四边形ABCD中,EF分别是边ABCD的中点,则ADBCEF的关系是AD+BC≥2EF

故选:B

5.(宛城区期中)如图,在△ABC中,∠A90°ACAB4,点DE分别在边ABAC上,BD4CE3,取DEBC的中点MN,线段MN的长为(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A2.5 B3 C4 D5

【分析】如图,作CHAB,连接DN,延长DNCHH,连接EH,首先证明CHBD,∠ECH90°,解直角三角形求出EH,利用三角形中位线定理即可解决问题.

【解答】解:作CHAB,连接DN并延长交CHH,连接EH

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BDCH

∴∠B=∠NCH,∠ECH+∠A180°

∵∠A90°

∴∠ECH=∠A90°

在△DNB和△HNC中,

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∴△DNB≌△HNCASA),

CHBD4DNNH

Rt△CEH中,CH4CE3

EH <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 5

DMMEDNNH

MN <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> EH2.5

故选:A

6.(凤山县期末)如图,在四边形ABCD中,∠A90°AB4MN分别是边BCAB上的动点(含端点,但点M不与点B重合)点EF分别是线段DMMN的中点,若线段EF的最大值为2.5,则AD的长为(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A5 B <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> C2.5 D3

【分析】根据三角形的中位线定理得出EF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> DN,从而可知DN最大时,EF的最大值为2.5,因为NB重合时DN最大,此时根据勾股定理求得DNDB

【解答】解:∵点EF分别是线段DMMN的中点,

EDEMMFFN

EF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> DN

DN最大时,EF最大,

线段EF的最大值为2.5

DN2EF5

NB重合时DN最大,

此时DNDB <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 5

AD3

故选:D

7.(鄞州区期末)如图,四边形ABCD中,∠B90°AB8BC6,点M是对角线AC的中点,点NAD边的中点,连结BMMN,若BM3MN,则线段CD的长是(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> B3 C <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> D5

【分析】首先由勾股定理求得AC的长度,结合直角三角形斜边上中线的性质得到BM <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AC,三角形中位线定理得到CD2MN

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 解答】解:如图,在直角△ABC中,∠B90°AB8BC6,则由勾股定理知,AC <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 10

NAD边的中点,

BM <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AC5

BM3MN

MN <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BM <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

M是对角线AC的中点,点NAD边的中点,

MN是△ACD的中位线.

CD2MN <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

故选:C

8.(陈仓区期末)如图所示,在四边形ABCD中,ABCD4MNP分别是ADBCBD的中点,∠ABD20°,∠BDC80°,则MN的长是  

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

【分析】作PHMNH,根据三角形中位线定理求出PMPN、∠MPN,根据等腰三角形的性质、勾股定理计算即可.

【解答】解:作PHMNH

MNP分别是ADBCBD的中点,

PM <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AB2PN <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> CD2PMABPNCD

∴∠MPD=∠ABD20°,∠BPN=∠BDC80°PMPN

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>MPN120°

PMPN

∴∠PMN30°MHHN

PH <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> PM1

由勾股定理得,MH <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

MN2MH2 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

故答案为:2 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

9.(垦利区期末)如图,在四边形ABDC中,EFGH分别为ABBCCDDA的中点,并且EFGH四点不共线.当AC6BD8时,四边形EFGH的周长是  

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

【分析】根据三角形中位线定理得到FGEHFGEH,根据平行四边形的判定定理和周长解答即可.

【解答】解:∵FG分别为BCCD的中点,

FG <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BD4FGBD

EH分别为ABDA的中点,

EH <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BD4EHBD

FGEHFGEH

四边形EFGH为平行四边形,

EFGH <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AC3

四边形EFGH的周长=3+3+4+414

故答案为:14

10.(商丘四模)如图,四边形ABCD中,点EF分别为ADBC的中点,延长FECD延长线于点G,交BA延长线于点H,若∠BHF与∠CGF互余,AB4CD6,则EF的长为  

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

【分析】根据三角形的中位线定理和勾股定理解答即可.

【解答】解:连接BD,取BD的中点M,连接EMFM

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> EF分别为ADBC的中点,MBD的中点,

EMMF分别为△ADB、△BCD的中位线,

EMABMFDCEM <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AB2MF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> DC3

MFDC

∴∠FGC=∠EFM

EMAB

∴∠FEM=∠FHB

∵∠BHF与∠CGF互余,

∴∠CGF+∠BHF=∠EFM+∠FEM90°

∴∠EMF180°﹣∠EFM﹣∠FEM90°

∴△EMF是直角三角形,

EF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

故答案为: <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

11.(莱州市期末)如图,两个等腰Rt△ABCRt△CEF,点BCE上,∠ABC=∠E90°,连接AF,取AF的中点M,连接MB.求证:BMCF

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

【分析】如图所示,延长ABCF于点D.根据全等三角形的性质得到ABBD,推出BM是△ADF的中位线,于是得到结论.

【解答】证明:如图所示,延长ABCF于点D

∵∠ABC90°

∴∠CBD90°

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> Rt△ABCRt△CEF是等腰直角三角形,

∴∠ACB=∠ECF45°

BCBC

∴△ACB≌△DCBASA),

ABBD

MAF的中点,

ANFM

BM是△ADF的中位线,

BMCF

类型三中位线的构造方法总结

(一). 连接两点构造三角形的中位线

如图,点BAC上一点,分别以ABBC为边在AC同侧作等边△ABD和等边△BCE,点PMN分别为ACADCE的中点.

1)求证:PMPN

2)求∠MPN的度数.

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

【分析】(1)连接DCAEAECD于点M,证明△ABE≌△DBC,得到AEDC,利用中位线的性质证明PMPN

2)根据中位线的性质把∠MPA+∠NPC转化成∠MCA+∠MAC,根据∠DMA=∠MCA+∠MAC可知求出∠DMA度数即可.

【解答】解:(1)连接DCAEAECD于点M

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>ABE和△DBC中,

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∴△ABE≌△DBCSAS).

AEDC

PAC中点,NEC中点,

PN <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AE

同理可得PM <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> DC

所以PMPN

2)∵PAC中点,NEC中点,

PNAE

∴∠NPC=∠EAC

同理可得∠MPA=∠DCA

∴∠MPA+∠NPC=∠EAC+∠DCA

又∠DQA=∠EAC+∠DCA

∴∠MPA+∠NPC=∠DQA

∵△ABE≌△DBC

∴∠QDB=∠BAQ

∴∠DQA=∠DBA60°

∴∠MPA+∠NPC60°

∴∠MPN180°﹣60°120°


  1. 利用角平分线和垂直构造中位线

1.(芝罘区期末)如图,在△ABC中,AB6AC4ADAE分别是角平分线和中线,过点CCFAD于点F,连接EF,则线段EF的长为(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A1 B2 C4 D <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

【分析】延长CFABG,根据等腰三角形的判定和性质得到AGAC4FGCF,进而求出BG,根据三角形中位线定理计算即可.

【解答】解:延长CFABG

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AD为△ABC的角平分线,CGAD

∴△ACG是等腰三角形,

AGAC4FGCF

BGABAG6﹣42

AE为△ABC的中线,

EF是△BCG的中位线,

EF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BG1

故选:A

2.(东宝区校级月考)在△ABC中,点DAB的中点,CE平分∠ACBAECE于点E

1)求证:DEBC

2)若AC5BC7,求DE的长.

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

【分析】(1)根据CE平分∠ACBAECE,运用ASA易证明△ACE≌△FCE.根据全等三角形的性质,得AEEFCFAC,根据三角形的中位线定理即可得到结论;

2)根据三角形的中位线定理就可求解.

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 解答】解:(1)延长AEBCF

CE平分∠ACBAECE于点E

∴∠ACE=∠FCE,∠AEC=∠FEC90°

在△ACE和△FCE中,

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∴△ACE≌△FCE

AEEF

DAB的中点,

ADBD

DE是△ABF的中位线.

DEBC

2)∵△ACE≌△FCE

CFAC5

DE是△ABF的中位线.

DE <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BCAC)= <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 7﹣5)=1

DE的长为1


  1. 倍长法构造三角形中位线

(越秀区校级二模)如图,△ABC、△BEF为等腰直角三角形,∠ABC=∠BEF90°BABCEBEF

连接AFCFMAF的中点.

1)如图1,当AFB共线时,求证:ME <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> CF

2)如图2,当AFB不共线时,求证:ME <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> CF

3)设BC2,请直接写出BF+AF+CF的最小值.

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

【解答】(1)证明:如图1中,延长FED,使EDEF,连接ADBD

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>BEF为等腰直角三角形,∠BEF90°

∴∠BFE45°BEDF

BE垂直平分DF

∴∠BDE45°

∴△BDF是等腰直角三角形,

BDBF,∠DBF90°

在△ABD和△CBF中,

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∴△ABD≌△CBFSAS),

ADCF

MAF的中点,DEEF

ME是△ADF的中位线,

ME <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AD

ME <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> CF


2)证明:如图2中,延长FED,使EDEF,连接ADBD

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>BEF为等腰直角三角形,∠BEF90°

∴∠BFE45°BEDF

BE垂直平分DF

∴∠BDE45°

∴△BDF是等腰直角三角形,

BDBF,∠DBF90°

∵∠CBF+∠ABF=∠ABC90°

ABD+∠ABF=∠DBF90°

∴∠CBF=∠ABD

在△ABD和△CBF中,

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∴△ABD≌△CBFSAS),

ADCF

MAF的中点,DEEF

ME是△ADF的中位线,

ME <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AD

ME <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> CF


 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 3)解:如图3中,以CF为边在CF的右侧作等边△CFM,将△CFB绕点C逆时针旋转60°得到△CME,连接AE,作EHACH,在EH上取一点D,使得CDDE,连接DC

CFFMFBME

AF+CF+FBAF+FM+ME

AEAF+FM+ME

AFME共线时,AF+FC+BF的值最小,

∵∠ACB45°,∠BCE60°

∴∠ACE45°+60°105°

∴∠ECH75°

∵∠H90°

∴∠CEH15°

DCDE

∴∠DCE=∠CED15°

∴∠CDH=∠DCE+∠DEC30°

CHa,则DCDE2aDH <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> aEH <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a+2a

Rt△ECH中,∵EC2CH2+EH2

22a2+ <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> a+2a2

a <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> (负根已经舍弃),

Rt△AEH中,AH2 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> + <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> EH <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

AE <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>  <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> + <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>


  1. 已知一边中点,取另一边中点构造三角形中位线

(衢州期末)如图,四边形ABCD的对角线ACBD相交于点OACBDEF分别是ABCD的中点,若ACBD2,则EF的长是(  )

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A2 B <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> C <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> D <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

【分析】取BC的中点GAD的中点H,连接EGGFFHHE,根据三角形中位线定理分别求出EGGF,得出四边形EGFH为正方形,根据正方形的性质计算即可.

【解答】解:取BC的中点GAD的中点H,连接EGGFFHHE

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> EG分别是ABBC的中点,AC2

EG <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> AC1EGAC

同理:FH <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ACFHACEG <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ACGFBDGF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BD1

四边形EGFH为平行四边形,

ACBD

GEGF

平行四边形EGFH为菱形,

ACBDEGACGFBD

EGGF

菱形EGFH为正方形,

EF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> EG <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

故选:D

  1. 已知两边中点,取第三边中点构造三角形的中位线

已知:如图所示,在△ABC中,ABACADBC边上的高,PAD的中点,延长BPAC于点F

1)求证:PB3PF

2)如果AC的长为13,求AF的长.

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

【分析】(1)本题可通过构建中位线来求解,过D点作DEBF,交ACE;则DEPF分别是△CBF、△ADE的中位线,可根据BPPFDE的比例关系求出BPPF的比例关系.

2)由(1)可知:EFAC的三等分点,由此可得出AF的长.

【解答】解:(1)证明:如图所示,过D点作DEBF,交ACE

因为ABACAD为△ABC的高,

所以根据等腰三角形的三线合一得DBC的中点,

所以DE <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BF

 <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 理,因为PAD的中点

所以PF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> DE,即PF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BF,所以BP3PF

2)由(1)得:PFDE分别是DEBF的中位线,

AFEFCEEF

ACAF+EF+CE3AF

AC13

AF <a href="/tags/14/" title="复习" class="c1" target="_blank">复习</a> <a href="/tags/38/" title="突破" class="c1" target="_blank">突破</a> <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/282/" title="专题" class="c1" target="_blank">专题</a> <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>




1