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【330009】15.2 线段的垂直平分线

时间:2025-02-07 09:17:21 作者: 字数:4878字



课后训练


1(如图,直线CD是线段AB的垂直平分线,P为直线CD上的一点,已知线段PA5,则线段PB的长度为(  )

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

(1题图)

A6     B5    C4    D3

2.已知MN是线段AB的垂直平分线,CDMN上的任意两点,则∠CAD与∠CBD之间的关系是(  )

A.∠CAD>∠CBD B.∠CAD=∠CBD C.∠CAD<∠CBD D.不能确定

3.如图,在平行四边形ABCD中,AB3BC5AC的垂直平分线交ADE,则△CDE的周长是(  )

A6 B8 C9 D10

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>  <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

(3题图) (4题图)

4.如图,△ABC中,AB的垂直平分线交ACD,如果AC5 cmBC4 cm,那么△DBC的周长是(  )


A6 cm B7 cm C8 cm D9 cm

5.如果三角形三条边的中垂线的交点在三角形的外部,那么这个三角形是(  )

A.直角三角形      B.锐角三角形

C.钝角三角形      D.等边三角形

6.已知直线MN是线段AB的垂直平分线,垂足为D,PMN上一点,AB10 cm,BD cm;若PA10 cm,则PB__________ cm.

7.如图,等腰三角形ABC中,已知ABAC,∠A30°AB的垂直平分线交ACD,则∠CBD的度数为________

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

(7题图)

8.如图,在RtABC中,∠C90°,∠B15°DEAB的垂直平分线,垂足为D,交BCEBE5,则AE______,∠AEC______.

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

(8题图)

9.如图,在△ABC中,BC边的垂直平分线DE交边BC于点D,交边AB于点E.若△EDC的周长为24,△ABC与四边形AEDC的周长之差为12,则线段DE的长为________

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

(9题图)

10.如图所示,在△ABC中,BC的垂直平分线EDAB于点E,交BC于点D,连接CE.求证:ABAC.

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

(10题图)

11.如图,在△ABC中,∠C90°,用直尺和圆规在AC上作点P,使点P到点AB的距离相等(保留作图痕迹,不用证明)

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

(11题图)

12.如图,P是∠AOB的平分线OM上任意一点,PEOAEPFOBF,连接EF.求证:OP垂直平分EF.

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

(12题图)

13.如图所示,已知OAOBACBD,且OAACOBBDMCD的中点.求证:(1)OM平分∠AOB(2)OMCD的垂直平分线.

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

(13题图)











答案与解析

1B 2B

3B 解析:利用线段的垂直平分线的性质可知,CDE的周长即为CDDEEC的长,CDDEECCDAD358.

4D 解析:△DBC的周长=BDCDBCADCDBCACBC549(cm)

5C

65 10

745° 解析:∵ABAC,∠A30°,∴∠ABC=∠C75°.AB的垂直平分线交ACD

ADBD.∴∠ABD=∠A30°.∴∠CBD=∠ABC-∠ABD75°30°45°.

85 30° 解析:线段垂直平分线上的点与线段两端距离相等,故AEBE.三角形的一个外角等于与它不相邻的两个内角的和,故∠AEC2B.

96 解析:∵DEBC边的垂直平分线,∴ECEB.∴△EDC的周长等于△BDE的周长,BDBEDE24.

又∵△ABC与四边形AEDC的周长之差为12,∴BDBEDE12.

∴①-②,得2DE12.DE6.

10.证明:∵EDBC的垂直平分线,

EBEC.

又∵在△AEC中,AEECAC

AEEBAC,即ABAC.

11.解:如图,分别以AB为圆心,以大于 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a> 的长为半径作弧分别交于MN两点,过MN作直线交AC于点P,则点P即为所求.


 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

(11题答图)

解析:在以A,B为圆心画弧时,一定要以大于 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a> 的长为半径作弧,否则由于两弧不相交而得不到交点.

12.证明:∵PEOAEPFOBF

∴∠PEO=∠PFO90°.

OM平分∠AOB,∴∠EOP=∠FOP.

在△PEO和△PFO中,

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

∴△PEO≌△PFO.(AAS)

PEPFEOFO.

OPEF的垂直平分线上.

OP垂直平分EF.


13.证明:(1)如图,连接OCOD.

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

(13题答图)

OAACOBBD,∴∠A=∠B90°.

在△AOC和△BOD中,

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

∴△AOC≌△BOD.(SAS)

∴∠AOC=∠BODOCOD.

MCD的中点,

CMDM.

在△CMO和△DMO中,

 <a href="/tags/75/" title="垂直" class="c1" target="_blank">垂直</a> <a href="/tags/175/" title="线段" class="c1" target="_blank">线段</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/907/" title="垂直平分线" class="c1" target="_blank">垂直平分线</a>

∴△CMO≌△DMO.(SSS)

∴∠COM=∠DOM.

∴∠AOC+∠COM=∠BOD+∠DOM

即∠AOM=∠BOM

OM平分∠AOB.

(2)(1),得△CMO≌△DMO

∴∠OMC=∠OMD90°

OMCD.

又∵MCD的中点,

OMCD的垂直平分线.