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【323281】2023八年级数学上册 第2章 三角形2.4 线段的垂直平分线2.4.2 线段垂直平分

时间:2025-01-15 20:28:53 作者: 字数:3282字
简介:


2课时 线段垂直平分线的作法



 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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.如图2420,在△ABC中,分别以点A和点B为圆心,大于AB的长为半径画弧,两弧相交于点M <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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> N,作直线MN,交BC于点D,连接AD.若△ADC的周长为10AB7,则△ABC的周长为 ( <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

2420

A7        <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

B14

C17  <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

D20

2.为了推进新型农村合作医疗制度改革,准备在某镇新建一个医疗点P,使P到该镇所属A村、B村、C村的村委会所在地的距离都相等(ABC不在同一直线上,地理位置如图2421所示),请你用尺规作 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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> 图的方法确定点P的位置.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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> 求:写出已知、求作;不写作法 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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> ,保留作图痕迹.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

2421








3.如图2422,一张纸上有线段AB.

(1)请用尺规作图,作出线段AB的垂直平分线(保留作图痕迹,不写作法和证明)

(2)若不用尺规作图,你还有其他作法吗?请说明作法(不作图)

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

2422









 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

4.如图2423,已知△ABC.

(1)BC边的垂直平分线交BCD,连接AD(尺规作图,不写作法,保留作图痕迹)

(2)(1)的基础上,若△ABC的面积为6,则△ABD的面积为________

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

2423

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

5[2012·凉山州]在学习轴对称的时候 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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> ,老师让同学们思考课本中的探究题.

如图2424(1),要在燃气管道l上修建一个泵站,分别向AB两镇供气,泵站修在管道的什么地方,可使所用的输气管线最短?

你可以在l上找几个点试一试,能发现什么规律?

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

2424

聪明的小华通过独立思考,很快得出了解决这个问题的正确办法.他把管道l看成一条直线(2424(2)),问题就转化为,要在直线l上找一点P,使APBP的和最小.他的作法是这样的:

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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> 作点B关于直线l的对称点B′.

连接AB交直线l于点P,则点P为所求.

请你参 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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> 考小华的作法解决下列问题.如图2425在△ABC中, <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>DE分别是ABAC边的中点,请你在BC边上确定一点P,使△PDE的周长最小.(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/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/387/" title="三角形" class="c1" target="_blank">三角形</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>

2425







答案解析

1C 【解析】 因为在△ABC中,分别以点A和点B为圆心,大于AB的长为半径画弧,两弧相交于点MN,作直线MN,交BC于点D,连接AD.所以M <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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> NAB的垂直平分线,所以ADBD,因为△ADC的周长为10,所以AC <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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> ADCDACBDCDACBC10,因为AB7,所以△ABC的周长为ACBCAB10717.故选C.

2解:已知:A村、B村、C村,

求作:新建一个医疗点P,使P <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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> 该镇所属A村、B村、C村的村委会所在地的距离都相等.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

2题答图


3解:(1)如图所示;

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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题答图

(2)对折,使得 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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与点B重合,则折痕所在的直线为线段AB的垂直平分线.

4解:(1)如图.

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

4题答图

(2)因为BDCD

所以SABDSADCSABC×63.

5解:D点关于BC的对称点D,连接DE,与BC交于点P,点P即为所求;

 <a href="/tags/55/" title="数学" class="c1" target="_blank">数学</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/387/" title="三角形" class="c1" target="_blank">三角形</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>

5题答图



5