名校
1 . 如图,在
中,D是
边的中点,过点D的直线交
于点E,交
的延长线于点F,且
.
上方作
交线段
于点G(用基本作图,保留作图痕迹,不写作法、不下结论)
(2)在(1)中所作的图中,证明:
(请补全下面的证明过程).
证明:
∵D为
边中点,
∴![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1b7513e09cccf40c3a483064b97e48.png)
∵![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01ab193c067c7db8757cce51b4524442.png)
∴ ① .
∴![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbcf0550a7a5ed75ee87d09d962cb60a.png)
在
和
中
.
∴
,
∴ ③ .
∵![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f805768a5ffaf8bdfa4bc3b680aafdc9.png)
∴
,
∴ ④ .
又∵
,
∴ ⑤ .
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f805768a5ffaf8bdfa4bc3b680aafdc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08f58ecbd98589db9188847a321ffe68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
(2)在(1)中所作的图中,证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d38d97f03faed3152db2fd3bd1919944.png)
证明:
∵D为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1b7513e09cccf40c3a483064b97e48.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01ab193c067c7db8757cce51b4524442.png)
∴ ① .
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbcf0550a7a5ed75ee87d09d962cb60a.png)
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcaecf08a22124a457128fb04c9c02bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f2281cb6df0c3c518ce5ed19a02b57e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52cc2fae2fedc1b43df2aa7cc1ef8364.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/765d1f35fa499080278d4e3f4dcd158e.png)
∴ ③ .
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f805768a5ffaf8bdfa4bc3b680aafdc9.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047eba5bba202f75349f94ad9feb4962.png)
∴ ④ .
又∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbcf0550a7a5ed75ee87d09d962cb60a.png)
∴ ⑤ .
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d38d97f03faed3152db2fd3bd1919944.png)
您最近一年使用:0次
2023-10-16更新
|
435次组卷
|
3卷引用:重庆市第八中学2023-2024学年九年级上学期第一次月考数学试题
重庆市第八中学2023-2024学年九年级上学期第一次月考数学试题重庆市第十一中学校2023-2024学年九年级上学期12月月考数学试题(已下线)中考重点01 尺规作图+补全证明过程(5题型+满分技巧+限时检测)-2024年中考数学【热点·重点·难点】专练(重庆专用)
2 . 如图,已知
.
(1)尺规作图:过点A作直线
(保留作图痕迹,不写作法);
(2)在l上截取
(点D在点A的右侧),连接
,线段
与
相于点O,
过点O且与线段
分别交于点E,F.请在(1)图中补全图形,并求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/16/3d5d233c-8623-486d-9b0f-5dda14c0c4c4.png?resizew=165)
(1)尺规作图:过点A作直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60a1eb9d682e2bd181f3b0f5115160b1.png)
(2)在l上截取
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d2530e7023b2345c651e8f53629ff1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83fe060a37a93ea736e9c3d444809cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcf2e3236ea30ee2c37928b98041f13a.png)
您最近一年使用:0次
名校
3 . 如图,已知矩形
,
,E为
延长线上一点,连接
交
于点F.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/19/2e02dfa7-40bd-4d52-9254-38e72eb4d3e8.png?resizew=145)
(1)尺规作图:过点B作
的垂线交
于点G.(保留作图痕迹,不写作法)
(2)在(1)所作的图形中,连接
,若
,求证:
平分
.为证明
平分
,小明的思路是将其转化成证明三角形全等,然后根据全等三角形的性质和角平分线的定义使问题得到解决.(请根据小明的思路补全下面的证明过程)
证明:四边形
是矩形,∴ ① ,∴
.
∵
,∴ ② ,∴
.
∵ ③ ,∴
.
∵在矩形
中,
,∴ ④ .
又∵ ⑤ ,∴
,∴ ⑥ ,∴
平分
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b9199a3de404cb13ae7710f73db5c8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/19/2e02dfa7-40bd-4d52-9254-38e72eb4d3e8.png?resizew=145)
(1)尺规作图:过点B作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
(2)在(1)所作的图形中,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14833dbeed409b33acd4c9071fd0be36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6a14654f44613e7dc85413f173e5cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6a14654f44613e7dc85413f173e5cd0.png)
证明:四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08613db7a612bb24a4a27e9886bd7def.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c7e48723871f06a6aeae31a2a1ff79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c097263313a866818aa456867f48c5f0.png)
∵ ③ ,∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9017dc1f04f6725bc6ee3fb631ed9619.png)
∵在矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3bbe4cdd2c154bd9a8073b0d4cecb8a.png)
又∵ ⑤ ,∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fea32039a4f78022f7d95561efc8ea3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6a14654f44613e7dc85413f173e5cd0.png)
您最近一年使用:0次
2023-05-15更新
|
373次组卷
|
3卷引用:重庆市云阳县第一初级中学教育集团2022-2023学年九年级下学期期中数学试题
名校
4 . 如图,矩形
的对角线
、
交于点
,
于
.
作
的垂线,垂足为
,连接
、
保留作图痕迹,不写作法,不写结论
.
(2)补全推理过程:
在矩形
中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
,
,
______ ,
,
,
,
即:______ ,
______ ;
在
和
中,
,
______ ,
四边形
为平行四边形
______
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52ab924e3692515bd8be4c36472a959a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9f035df1355a1ad3b1734b1f4c9dbee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04582116cd765fcc5a52f44279ad6c94.png)
(2)补全推理过程:
在矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4adf90a8c2b29334cdc5aa5b554991f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d2530e7023b2345c651e8f53629ff1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df268f41c00277f007f09dcdfc37bacf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4c8f0f573021b130dd15677f3ed96d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95ed53ad9bfff65c2b12b55d9d0baae0.png)
即:______ ,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7f9fba8a4098c1a0515286eb8d616dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e817a54942ebf021a2967ff0c51a8111.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d86a5408d00808362b153a328cdd6cdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d58c20643bee53ffb619bda15f8c4c35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b977f9880c3acc912c36aa3e2df56c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd995178601c2ad7b40f973d268c7bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/913b7537e011acfeec11952731351388.png)
您最近一年使用:0次
2023-06-13更新
|
386次组卷
|
3卷引用:重庆市江津实验中学、京师实验学校等金砖四校2022-2023学年八年级下学期期中数学试题
5 . 数学课上,老师出示了一道题目:如图1,在
中,
,点E在
上,点D在
的延长线上,且
,试探究线段
之间存在的数量关系,并说明理由.
(1)[猜想证明]线段
的关系是
.请补全下列证明思路;
如图1:过点E作
交
于点F,则
,
∵
,
∴
,
∴
,
又∵
,
∴
,
∵
.
∴
.
∴
(ASA),
∴
,
∴
.
∴
,
∴
,
∴
.
(2)[变式拓展]
如图2,在
中,
,点E在
的延长线上,点D在直线
上,且
.请你在图2中补齐图形.并探索(1)中的结论是否仍然成立?若成立,请给出完整的证明;若不成立,请直接写出新的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62f9a3935cd124d021c7b18b0f634915.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50e0138b41ea8fd06e5c2da507369464.png)
(1)[猜想证明]线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50e0138b41ea8fd06e5c2da507369464.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccbfe87d0b106dcdaa99489b88172209.png)
如图1:过点E作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee73452ee4d5437f1399f1235b95e55f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba7ee0623e5c5cb5917da7fe3fd92054.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62f9a3935cd124d021c7b18b0f634915.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb8e46e5413602e83c6edb272b03194c.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eecf2f42df5ba545893d84939c8a3b6.png)
又∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b77c8ef75793d21d2d5d8bf470a61159.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0913d30bc3495feae31235f6a2b3ff1e.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aa84f9a2c40c9e0141be50c42fd114a.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b31772ed229bd95f9e47196abcd273b0.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9332b9191d830338ed51f96422879f03.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/306f134e6b56cb208d550f54f4bbc6be.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b3dcd9161991c027aed30b55f3b7eda.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b3dcd9161991c027aed30b55f3b7eda.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccbfe87d0b106dcdaa99489b88172209.png)
(2)[变式拓展]
如图2,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71c472fb321d44281f1e6c4da4ecff6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dea2ae9d515f9ab351ad72306b776ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62f9a3935cd124d021c7b18b0f634915.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/4/22f2e61f-2f4c-4fa4-a89d-9bd4de3b47af.png?resizew=298)
您最近一年使用:0次
名校
6 . 已知四边形ABCD为矩形(AD>AB),
![](https://img.xkw.com/dksih/QBM/2022/3/28/2945929342631936/2946524035260416/STEM/1567314a-e6a5-4496-8efc-6529a0537933.png?resizew=243)
(1)尺规作图:在BC上取一点E,使AE=AD;过点D作DF⊥AE,交AE于点F(基本作图,保留作图痕迹,不写作法,);
(2)求证:DF=DC,(请补全下面的证明过程,不写证明理由)
证明:∵______①,
∴AD=BC,AD∥BC,∠B=90°,
∴∠FAD=∠AEB.
∵DF⊥AE,
∴______②.
∴______③.
在△AFD与△EBA中,
,
∴△AFD≌△EBA(AAS).
∴______④.
又∵AB=CD,
∴DF=DC.
![](https://img.xkw.com/dksih/QBM/2022/3/28/2945929342631936/2946524035260416/STEM/1567314a-e6a5-4496-8efc-6529a0537933.png?resizew=243)
(1)尺规作图:在BC上取一点E,使AE=AD;过点D作DF⊥AE,交AE于点F(基本作图,保留作图痕迹,不写作法,);
(2)求证:DF=DC,(请补全下面的证明过程,不写证明理由)
证明:∵______①,
∴AD=BC,AD∥BC,∠B=90°,
∴∠FAD=∠AEB.
∵DF⊥AE,
∴______②.
∴______③.
在△AFD与△EBA中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e66cd885e91cf8acbafb9db98c412010.png)
∴△AFD≌△EBA(AAS).
∴______④.
又∵AB=CD,
∴DF=DC.
您最近一年使用:0次
2022-03-29更新
|
704次组卷
|
3卷引用:重庆市第八中学2021-2022学年九年级下学期第二次月考数学试题
7 . 如图,在
中,
,
,
于点D(直线
不与
边相交).
![](https://img.xkw.com/dksih/QBM/2021/7/23/2770471612989440/2777544514519040/STEM/d4270b76-4066-4e72-bf05-e9ff1afbebfd.png?resizew=198)
(1)在所给图形中过点C作
于点E;(要求:尺规作图,保留作图痕迹,并用
黑色字迹笔描重作图痕迹,不必写作法和证明)
(2)求证:
.(说明:如果尺规作图没有完成,可直接画出草图进行证明)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c06154cae3bf7a8ce5a1e97a7380875.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93380ab8eccc6e6f11bdf19ffc2d4cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://img.xkw.com/dksih/QBM/2021/7/23/2770471612989440/2777544514519040/STEM/d4270b76-4066-4e72-bf05-e9ff1afbebfd.png?resizew=198)
(1)在所给图形中过点C作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/497846628a41a9bc750a645e045afb47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f502bc483fc90d900d8261d88cab8ae.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb773a664fa72fc5d8fe377e9f891901.png)
您最近一年使用:0次
8 . 戴高乐是二战期间领导法国人民赶走德国法西斯的英雄,也是法兰西第五共和国的总统.他去世后,根据他生前的意愿,他的墓前只立有一块小小的碑牌,一面刻着“查尔斯·戴高乐1890—1970”,另一面则刻着一个洛林十字架.洛林十字架由13块相同的小正方形组成,如图1所示.
![](https://img.xkw.com/dksih/QBM/2022/4/15/2958595551428608/2962421417205760/STEM/4f7f9aa0-ade4-430e-af69-a00d94ba92f3.png?resizew=477)
(1)你能否只用一把无刻度直尺画一条直线,使其等分洛林十字架.(面积等分,在图1中画出1种情形即可)
(2)戴高乐还是第一个提出并且解决了下面一个非常有趣的有关洛林十字架的数学问题的人.问题如下:如图2,在洛林十字架的A点处作一条直线,把洛林十字架严格地划分成面积相等的两部分.
戴高乐利用圆规,直尺和铅笔解决了该问题,他的作法如下:如图3所示,①标记点D,B,M,连接BM,与AD交于点F;②以点F为圆心,FD长为半径作弧,与BF交于点G;③以点B为圆心,BG长为半径作弧,与BD交于点C;④连接CA并延长,与洛林十字架边界交于点N,则直线CN即为所求.
请根据戴高乐的作图步骤,证明直线CN等分洛林十字架.小林同学的部分证明过程如下:
标记点H,P,Q,如图3所示.设洛林十字架中每个小正方形的边长为1.
易证
,
∴
.
由作图,可知
.
∴
.
∴
.
∴
.
请补全小林同学的证明过程.
![](https://img.xkw.com/dksih/QBM/2022/4/15/2958595551428608/2962421417205760/STEM/4f7f9aa0-ade4-430e-af69-a00d94ba92f3.png?resizew=477)
(1)你能否只用一把无刻度直尺画一条直线,使其等分洛林十字架.(面积等分,在图1中画出1种情形即可)
(2)戴高乐还是第一个提出并且解决了下面一个非常有趣的有关洛林十字架的数学问题的人.问题如下:如图2,在洛林十字架的A点处作一条直线,把洛林十字架严格地划分成面积相等的两部分.
戴高乐利用圆规,直尺和铅笔解决了该问题,他的作法如下:如图3所示,①标记点D,B,M,连接BM,与AD交于点F;②以点F为圆心,FD长为半径作弧,与BF交于点G;③以点B为圆心,BG长为半径作弧,与BD交于点C;④连接CA并延长,与洛林十字架边界交于点N,则直线CN即为所求.
请根据戴高乐的作图步骤,证明直线CN等分洛林十字架.小林同学的部分证明过程如下:
标记点H,P,Q,如图3所示.设洛林十字架中每个小正方形的边长为1.
易证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/686a562facf1b6b11759266414ee8814.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/240dfdbfd3a0834a2af02916796dbe13.png)
由作图,可知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12051911eea77b8d0391265a738da164.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/271474eee1c02680028ad238f73cbfbc.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7389941e301e854f8581ef51230c68c9.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d49a9716222c243abd96dfd3de85cb8.png)
请补全小林同学的证明过程.
您最近一年使用:0次
名校
9 . 如图,在
中,点E在边
上,连接
.
求作一点F,使得
;(保留作图痕迹,不写作法)
(2)若
,证明:四边形
为菱形.
证明:∵四边形
是平行四边形,
∴______________________
,
.
∵
,
∴
(
).
∴
,______________________.
∵
,
∴
,
∴______________________
∵
,
∴四边形
是平行四边形(______________________).(填推理依据)
∵
,
∴四边形
是菱形(______________________).(填推理依据)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/754f59660f49c228ae111e0357fb7417.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98de02d1d5b7ac04bce54be393218922.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/910936ec9fb419d51ce2f5ea817f8401.png)
证明:∵四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
∴______________________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c220eadc312101e2fb89dfe920f7b30d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc9f07a876244afec67a0eca4d45c2e6.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5e1dc3473ec67beb52aaaba5645cacb.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e26dba530ba1c15d3d3d5f90e0bf2810.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4351a730f61bb998bab8f0b7848912d7.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ce1475f537b4ad21775bfaa16daa0c.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc9f07a876244afec67a0eca4d45c2e6.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f73f57e2287c50d910e3946c56530dca.png)
∴______________________
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ce1475f537b4ad21775bfaa16daa0c.png)
∴四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9766e5eb6796dafc5ffe212afdfc43c0.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98de02d1d5b7ac04bce54be393218922.png)
∴四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9766e5eb6796dafc5ffe212afdfc43c0.png)
您最近一年使用:0次
10 . 如图,在
中,D是
上一点.
上确定一点O,使得
(尺规作图,保留作图痕迹,不写作法);
(2)在(1)的条件下,当
时,将
绕点O旋转得到
,其中,D,E分别是点A,B的对应点,若D是
的中点,
交
于点G,求证:G是
的中点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c35c43d90814396ff06336fa45e9650.png)
(2)在(1)的条件下,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfd4678832abebbe835f3032d11f4e49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
您最近一年使用:0次