【问题情境】如图①,在△ABC中,若AB=10,AC=6,求BC边上的中线AD的取值范围.
(1)【问题解决】延长AD到点E使DE=AD,再连接BE(或将△ACD绕着点D逆时针旋转180°得到△EBD),把AB、AC、2AD集中在△ABE中,利用三角形三边的关系即可判断出中线AD的取值范围是 .
【反思感悟】解题时,条件中若出现“中点”、“中线”字样,可以考虑构造以该中点为对称中心的中心对称图形,把分散的已知条件和所求证的结论集中到同个三角形中,从而解决问题.
(2)【尝试应用】如图②,△ABC中,∠BAC=90°,AD是BC边上的中线,试猜想线段AB,AC,AD之间的数量关系,并说明理由.
(3)【拓展延伸】如图③,△ABC中,∠BAC=90°,D是BC的中点,DM⊥DN,DM交AB于点M,DN交AC于点N,连接MN.当BM=4,MN=5,AC=6时,请直接写出中线AD的长.
(1)【问题解决】延长AD到点E使DE=AD,再连接BE(或将△ACD绕着点D逆时针旋转180°得到△EBD),把AB、AC、2AD集中在△ABE中,利用三角形三边的关系即可判断出中线AD的取值范围是 .
【反思感悟】解题时,条件中若出现“中点”、“中线”字样,可以考虑构造以该中点为对称中心的中心对称图形,把分散的已知条件和所求证的结论集中到同个三角形中,从而解决问题.
(2)【尝试应用】如图②,△ABC中,∠BAC=90°,AD是BC边上的中线,试猜想线段AB,AC,AD之间的数量关系,并说明理由.
(3)【拓展延伸】如图③,△ABC中,∠BAC=90°,D是BC的中点,DM⊥DN,DM交AB于点M,DN交AC于点N,连接MN.当BM=4,MN=5,AC=6时,请直接写出中线AD的长.
![](https://img.xkw.com/dksih/QBM/2019/12/22/2360661092671488/2361303537352704/STEM/b8304d6d648f469bb4c2b2f5723afc40.png?resizew=519)
更新时间:2019-12-23 13:47:15
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解答题-作图题
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【推荐1】在
中,
,
,
为等边三角形,
,连接
,
为
中点.
(1)如图1,当
,
,
三点共线时,请画出
关于点
的中心对称图形,判断
与
的位置关系是 ;
(2)如图2,当A,
,
三点共线时,问(1)中结论是否成立,若成立,给出证明,若不成立,请说明理由;
(3)如图2,取
中点
,连
,将
绕点
旋转,直接写出旋转过程中线段
的取值范围是 .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faed94a64b2dcfc6801b4fca0f16675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d777adf392866d76f7b60ef508140d98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e918b70b02a73685e3c536c7f380e2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc4b17ce6e90cd3810a3696262e94c1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/833cfda415649b832cc136caed392753.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
(1)如图1,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50217f77f46d3dc6e84ee9441249d64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce6c0e9de83f2e64ae33609fc08459d.png)
(2)如图2,当A,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(3)如图2,取
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://img.xkw.com/dksih/QBM/2019/12/12/2353465503907840/2356996921024512/STEM/74a4134c32254805a1d38fab50b4cdf9.png?resizew=332)
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解答题-问答题
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【推荐2】【阅读理解】
课外兴趣小组活动时,老师提出了如下问题:
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/25/dc1b8682-8c90-479a-a3df-fc4cc3527c3b.png?resizew=312)
如图1,
中,若
,
,求
边上的中线
的取值范围.小明在组内经过合作交流,得到了如下的解决方法:延长
到点
,使
,请根据小明的方法思考:
(1)由已知和作图能得到
的理由是 .
.
.
.
.![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5eb2decaa6be2df36a5e4b7fabf585d.png)
(2)求得
的取值范围是 .
.
.
.
.![](https://staticzujuan.xkw.com/quesimg/Upload/formula/142869a7cc5e90b551b38210deff9f0f.png)
【感悟】
解题时,条件中若出现“中点”“中线”字样,可以考虑延长中线构造全等三角形,把分散的已知条件和所求证的结论集合到同一个三角形中.
【问题解决】
(3)如图2,
是
的中线,
交
于
,交
于
,且
.求证:
.
课外兴趣小组活动时,老师提出了如下问题:
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/25/dc1b8682-8c90-479a-a3df-fc4cc3527c3b.png?resizew=312)
如图1,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54275b7e571660d0a9e0370fbfe5050b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1682d306c38087d9e6f7efb9cec596a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ad98ad714864041a632ca949308e417.png)
(1)由已知和作图能得到
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2254b62536e7f96b5a529a26cab7251d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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(2)求得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145a085312b8fa4af76b85fd7669413.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab7f502effec463bec4889245c4f09a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/142869a7cc5e90b551b38210deff9f0f.png)
【感悟】
解题时,条件中若出现“中点”“中线”字样,可以考虑延长中线构造全等三角形,把分散的已知条件和所求证的结论集合到同一个三角形中.
【问题解决】
(3)如图2,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
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前进10
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坡度
(注:
,垂足为E,
),在点B处测得建筑物顶部A的仰角为53°,求
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,
,
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