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【330720】第一章 三角形的证明 周周测8(1.4)

时间:2025-02-11 18:39:30 作者: 字数:12045字

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> ·

1.4角平分线

一、选择题

1.到三角形三条边的距离都相等的点是这个三角形的(  )

A. 三条中线的交点       <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> B. 三条高的交点       <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> C. 三条边的垂直平分线的交点       <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> D. 三条角平分线的交点

2.如图,在ABC中, <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> 90°AE平分 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> CE6,则点EAB的距离是(    
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A. 8                                            <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> B. 7                                            <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> C. 6                                            <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> D. 5

3.如图,ABC的外角CBDBCE的平分线相交于点F,则下列结论正确的是(  )
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A. FBC边的垂直平分线上                                  <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> B. FBAC的平分线上
C. BCF是等腰三角形                                              <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> D. BCF是直角三角形

4.ABC中,C=90°AD为角平分线,BC=32BDDC=97,则点DAB的距离为( 

A. 18cm                                  B. 16cm                                  C. 14cm                                  D. 12cm

5.如图,直线abc表示交叉的公路,现要建一货物中转站,要求它到三条公路的距离相等,则可供选择的站址有(  )
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A. 一处                                     B. 两处                                     C. 三处                                     D. 四处

6.如图,OP平分AOBPDOAPEOB,垂足分别为DE,下列结论正确的是( 
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A. PD=PE                           B. PE=OE                           C. DPO=EOP                           D. PD=OD

7.如图,已知在ABC中,CDAB边上的高线,BE平分ABC,交CD于点EBC=5,DE=2,则BCE的面积等于(  
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A. 5                                           B. 7                                           C. 10                                           D. 3

8.如图,在ABC中,BAC的平分线AD=10AC=8,CD=6,则点DAB边的距离是(      
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A. 8                                        <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> B. 7                                        <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> C. 6                                        <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> D. 无法确定

9.如图,AOP=BOP=15°PC//OAPDOA,若PC=4,则PD等于(    
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A. 4                                            <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> B. 3                                            <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> C. 2                                            <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> D. 1

10.如图,ADABC的角平分线,DEAB于点ESABC=8ED=2AC=3,则AB的长是(  )
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A. 5                                            <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> B. 6                                            <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> C. 7                                            <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> D. 8

11.如图,在RtABC中,C=90°AD平分BAC,交BCD,若CD= <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> BD,点D到边AB的距离为6,则BC的长是(  )
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

A. 6                                          <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> B. 12                                          <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> C. 18                                          <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> D. 24

二、填空题

12.如图,在ABC中,C=90°AD是角平分线,AC=5DC=3,则点DAB的距离是________

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

13.如图,OCAOB的平分线,POC上一点,PDOA于点DPD=6,则点P到边OB的距离为________

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

14.如图,在RtABC中,C=90°AB=8AD平分BAC,交BC边于点D,若CD=2,则ABD的面积为________ 
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

15.如图,在ABC中,ACB=90°AD平分BAC BC=10cmBDDC=3:2,则点DAB的距离________cm
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

16.如图所示,在RtABC中,A=90°BD平分ABC,交AC于点D,且AD=2BC=5,则BCD的面积是________
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

17.如图,在ABC中,C=90°BD平分ABC,若BD=5BC=4,则点D到边AB的距离为________

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

18.OABC内一点,且点O到三边的距离相等,A=70°,则BOC的度数为________

19.表示三条相互交叉直线工路上,现要建一个货运中转站,要求它到三条公路的距离相等,则选择的地址有________处.

20.如图,在RtABC中,C=90°,以顶点A为圆心,适当长为半径画弧,分别交ACAB于点MN,再分别以点MN为圆心,大于  <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a> MN的长为半径画弧,两弧交于点P,作射线AP交边BC于点D,若CD=4AB=15,则ABD的面积是________

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

21.如图,在ABC中,C=90°AD平分BACDEABE,有下列结论:①CD=ED;②AC+BE=AB;③BDE=BAC;④DA平分CDE;⑤SABDSACD=ABAC.其中,正确的有________个.  <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

三、解答题

22.如图,已知PBABPCAC,且PB=PCDAP上的一点,求证:BD=CD

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>




23.现要在三角地ABC内建一中心医院,使医院到AB两个居民小区的距离相等,并且到公路ABAC的距离也相等,请确定这个中心医院的位置.
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

24.如图,BE=CFDEAB的延长线于点EDFAC于点F,且DB=DC
求证:ADBAC的平分线.
 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>







25.如图,在ABC中,AD平分BAC

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

1)求证:SABDSACD=ABAC

2)若AB=4AC=5BC=6,求BD的长.














答案:

  1. A 2.C 3.B 4.C 5.D 6.A 7.A 8.C 9.C

10.A 11.C 12.3 13.6 14.8 15.4 16.5 17.3

18.125° 19.4 20.30 21.5

22.证明:∵PB⊥BAPC⊥CA
Rt△PABRt△PAC

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

Rt△PAB≌Rt△PACHL
∴∠APB=∠APC
在△PBD与△PCD

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

∴△PBD≌△PCDSAS
∴BD=CD


23.解:

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>
AB的垂直平分线EF,作∠BAC的角平分线AM,两线交于P
P为这个中心医院的位置.


24.证明:∵DE⊥AB的延长线于点EDF⊥AC于点F
∴∠BED=∠CFD
∴△BDE与△CDF是直角三角形,

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

Rt△BDE≌Rt△CDF
∴DE=DF
∴AD是∠BAC的平分线.


25.1)证明:过DDE⊥ABEDF⊥ACF

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>
∵AD平分∠BAC
∴DE=DF

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

2解:∵SABDSACD=ABAC=BDCD

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>

BC=6

 <a href="/tags/387/" title="三角形" class="c1" target="_blank">三角形</a> <a href="/tags/886/" title="证明" class="c1" target="_blank">证明</a> <a href="/tags/920/" title="三角" class="c1" target="_blank">三角</a>