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【325286】河北省2024八年级数学上册 第十六章 轴对称和中心对称 16.3角的平分线预学检验+

时间:2025-01-15 22:20:21 作者: 字数:3063字
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第十六章 轴对称和中心对称

16.3 角的平分线

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1(2024孝感期末)如果一个角是75°,那么这个角的余角的度数是________

2.如图,ADAE分别是△ABC和△ABD的角平分线,且∠BAC60°,则∠EAC________

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

3.如图,分别过点P作∠AOB的两边OAOB的垂线.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

1.角平分线的性质定理:角平分线上的点到这个角的两边的________相等.

2.角平分线性质定理的逆定理:到角的两边________相等的点在角平分线上.

3(2024长春期末)如图,在四边形ABCD中,∠A90°,对角线BD平分∠ABC.AD5BC10,则△BCD的面积为(  )

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

A15 B20 C25 D50

4(2023珠海期中)如图,在直角三角形ABC中,∠C90°,请用尺规作图法在AC边上作一点P,使得点P到边ABBC的距离相等.(保留作图痕迹,不写作法)

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>







 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

知识点1 角平分线的性质定理

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 如图,在直角三角形ABC中,∠ACB90°BD平分∠ABCDEAB于点E.AC11DE5,则AD的长为________

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>


变式1如图,在△ABC中,∠ACB90°AD平分∠BACBC边于点D,△ABD的面积为30AB15,则线段CD的长度为________

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

知识点2 角平分线性质定理的逆定理

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 如图,在四边形ABCD中,ADBCABADCE平分∠DCBAB于点E,点EAB的中点.求证:点E在∠ADC的平分线上.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>


变式2如图,已知FGOA上两点,MNOB上两点,且FGMNSPFGSPMN,试问:点P是否在∠AOB的平分线上?请说明理由.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>



                                    

知识点3 作角的平分线

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 电信部门要修建一座电视信号发射塔P,按照设计要求,发射塔P到两城镇AB的距离必须相等,到两条高速公路mn的距离也必须相等.请在图中作出发射塔P的位置.(尺规作图,不写作法,保留作图痕迹)

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>




变式3(2024黄冈期中)如图,在直角三角形ABC中,∠C90°,用尺规作图法作出射线AEAEBC于点DCD2PAB上一动点,则PD的最小值为________

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

第十六章 轴对称和中心对称

16.3 角的平分线

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

115° 2.45°

3.解:如图所示.

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 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

1.距离 2.距离 3.C

4.解:如图所示,点P即为所求.

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16 变式1.4

2证明:过点EEFCD于点F

ADBCABAD,∴ABBC.

CE平分∠DCBABBCEFCD,∴EFBE.

EAB的中点,∴EAEB,∴EFEA.

又∵EAADEFCD,∴点E在∠ADC的平分线上.

变式2.解:点P在∠AOB的平分线上.

理由:过点P分别向OAOB作垂线,垂足为EH.

SPFGFG·PESPMNMN·PHFGMNSPFGSPMN

PEPH.∴P在∠AOB的平分线上.

3解:如图,点PP即为所求作.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</a> <a href="/tags/249/" title="对称" class="c1" target="_blank">对称</a> <a href="/tags/553/" title="河北" class="c1" target="_blank">河北</a> <a href="/tags/849/" title="轴对称" class="c1" target="_blank">轴对称</a> <a href="/tags/893/" title="河北省" class="c1" target="_blank">河北省</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

变式3.2