当前位置:首页 > 八年级 > 数学试卷

【329951】12.3.2角平分线的性质(2)

时间:2025-02-07 09:26:25 作者: 字数:3641字

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

12.3.2 角平分线的性质(2

学习目标

1、会叙述角平分线的性质及“到角的两边距离相等的点在角的平分线上”

2、能利用两个性质解决一些实际问题

学习重点角平分线的性质及应用

学习难点利用两个性质解决一些实际问题

课前预习

阅读课本,完成下列的问题:角平分线的判定及几何语言表

复习旧知——角平分线的性质定理

1、性质定理:角平分线上的点到角的 的距离

2、几何语言:(注意:三个已知条件缺一不可)

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>  <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

3画出三角形三个内角的平分线

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>  <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

你发现了什么特点吗?

4、如图,△ABC的角平分线BMCN相交于点P,求证,点P到三边ABBCCA的距离相等。

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>





课内探究

1、求证:到角的两边的距离相等的点在角的平分线上。

(提示:先画图,并写出已知、求证,再加以证明)

小帅已经做出了一些步骤,请你帮他补充完整:

解:如图,过______射线,

已知: <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> ;并且_______=_______

求证:_____ <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的平分线

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>


证明:




结论:角平分线的判定定理:

角的内部到角的两边____的点在角的___上。

注意:(1)该定理也是证明两角相等的一种方法;

2)三角形的三条角平分线交于一点,这点是三角形的内心,到三边的距离相等.

(3)符号语言:∵ PD⊥OAPE⊥OBPD= PE

______( )

4)作用:常证明两个角相等

2、比较角平分线的性质与判定

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

2如图,CDABBEAC,垂足分别为DEBECD相交于点OOBOC,求证∠1=∠2

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>







拓展延伸

如图,在四边形ABCD中,BC>BAAD=DC,BD平分∠ABC,求证:∠A+∠C=180°

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>



当堂检测

1如图,在四边形 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 中, <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>  <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 平分 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> ,且 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> ,求证: <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 平分 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>




2如图,在△ABC中,BD为∠ABC的平分线,DEAB于点E,且DE2cm AB9cmBC6cm,求△ABC的面积.

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>


课后训练

基础知识

1、已知△ABC中,A=60°,∠ABC,∠ACB的平分线交于点O,则∠BOC的度数为

2、到三角形三条边的距离相等的点是( )

A、三条中线的交点 B、三条高线的交点

C、三条边的垂直平分线的交点 D、三条角平分线的交点

3、下面哪个点到三角形三边的距离相等( )

A三条角平分线的交点 B三条中线的交点

C三条高的交点 D三角形内任意一点

4、如图, <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的两个外角平分线相交于点 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> ,则下面结论正确的是( )

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>


A <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 不平分 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> B <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 平分 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> C <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 平分 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> D <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

5、在 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 中, <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的角平分线,若 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> ,则点 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的距离为     

6、如图, <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的三边 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的长分别为203040,其三条角平分线的交点为 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> ,则 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>


 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

7 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的平分线上一点 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的距离为 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> ,则 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的距离为    <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>

8、如图,在直线 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 上求一点 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> ,使得点 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 到射线 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的距离相等.

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>


9、如图,在 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 中, <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> ,点 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 为三条角平分线的交点, <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> ,且 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> ,求 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的长.

 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>




10、如图, <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 内一点, <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 上, <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 上,且 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 的面积相等.

求证: <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a> 平分 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>


 <a href="/tags/386/" title="性质" class="c1" target="_blank">性质</a> <a href="/tags/904/" title="平分" class="c1" target="_blank">平分</a>