如图,在矩形
中,
为对角线,点
为
边上一动点,连结
,过点
作
,垂足为
,连结
.
(1)证明:
;
(2)当点
为
的中点时,若
,求
的度数;
(3)当点
运动到与点
重合时,延长
交
于点
,若
,则
.
![](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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0cfc81f5df5b164223424a6b2370714.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f53330c107f8245290a5a42c3d356acd.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f71d26492981b79653b325f2d9685965.png)
(2)当点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e25b0d3295d113384b806bebf39cb472.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3034c97c7553d154da59ee6ffb65cc4.png)
(3)当点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb6ede9761b5b90f8dc137708e1ee90f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/756e43777caa291cb950f4dd97bb0a97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d065cd1a9956ea5bafb58ae1ce466774.png)
![](https://img.xkw.com/dksih/QBM/2019/7/30/2258120330158080/2258932538974208/STEM/2910812b029d48168c615ef65fe6f2ba.png?resizew=218)
更新时间:2019-07-31 22:32:55
|
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解答题-证明题
|
较难
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【推荐1】探究活动一:
如图1,正方形ABCD和正方形QMNP,∠M=∠B,M是正方形ABCD的对称中心,MN交AB于F,QM交AD于E,线段ME与线段MF的数量关系是 .(不必证明,直接给出结论即可)
探究活动二:
如图2,将上题中的“正方形”改为“矩形”,且AB=mBC,其他条件不变(矩形ABCD和矩形QMNP,∠M=∠B,M是矩形ABCD的对称中心,MN交AB于F,QM交AD于E),探究并证明线段ME与线段MF的数量关系;
探究活动三:
根据前面的探索和图3,平行四边形ABCD和平行四边形QMNP中,若AB=mBC,∠M=∠B,M是平行四边形ABCD的对称中心,MN交AB于F,QM交AD于E,请探究并证明线段ME与线段MF的数量关系.
如图1,正方形ABCD和正方形QMNP,∠M=∠B,M是正方形ABCD的对称中心,MN交AB于F,QM交AD于E,线段ME与线段MF的数量关系是 .(不必证明,直接给出结论即可)
探究活动二:
如图2,将上题中的“正方形”改为“矩形”,且AB=mBC,其他条件不变(矩形ABCD和矩形QMNP,∠M=∠B,M是矩形ABCD的对称中心,MN交AB于F,QM交AD于E),探究并证明线段ME与线段MF的数量关系;
探究活动三:
根据前面的探索和图3,平行四边形ABCD和平行四边形QMNP中,若AB=mBC,∠M=∠B,M是平行四边形ABCD的对称中心,MN交AB于F,QM交AD于E,请探究并证明线段ME与线段MF的数量关系.
![](https://img.xkw.com/dksih/QBM/2018/5/22/1950748520423424/2008944738656256/STEM/5969c24230714301b0bf870d18e29c6f.png?resizew=514)
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【推荐2】如图,等边△ABC中,点D、E分别在边BC、AC上,AE=CD,连接AD、BE交于点P.
(1)求证:∠BPD=60°.
(2)连接PC,若CP⊥PB.当AP=3,求BP的长.
(1)求证:∠BPD=60°.
(2)连接PC,若CP⊥PB.当AP=3,求BP的长.
![](https://img.xkw.com/dksih/QBM/2020/2/5/2392275717496832/2392642917498880/STEM/93bc8b4e46ef4e67892247a6cd0ef96c.png?resizew=124)
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【推荐1】已知:如图①,在矩形ABCD中,
,垂足是E,点F是点关于AB的对称点,连接AF、BF.
![](https://img.xkw.com/dksih/QBM/2021/6/10/2740207511388160/2740668974637056/STEM/80f23923-a2b6-4c8d-96fd-c955d2c886ed.png)
(1)直接求出:
__;
__;
(2)若将
沿着射线BD方向平移,设平移的距离为
(平移距离指点B沿BD方向所经过的线段长度),点F分别平移到线段AB、AD上时,求出相应的m的值.
(3)如图②,将
绕点B顺时针旋转一个角
,记旋转中的
为
,在旋转过程中,设
所在的直线与直线AD交于点P,与直线BD交于点
是否存在这样的P、Q两点,使
为等腰三角形?若存在,直接写出此时DQ的长;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8453cd20043adcb9c91c0be1aa747869.png)
![](https://img.xkw.com/dksih/QBM/2021/6/10/2740207511388160/2740668974637056/STEM/80f23923-a2b6-4c8d-96fd-c955d2c886ed.png)
(1)直接求出:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e5f441402ee632377b6bedeb060f3d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4f76e37d6e6b182896ccaaa187cfe07.png)
(2)若将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/004104bafb5f30338123d4ea2b7fedde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)如图②,将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/004104bafb5f30338123d4ea2b7fedde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05df5a26c2b978503e93efa040b99508.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/004104bafb5f30338123d4ea2b7fedde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1107d067955bdc987e77291b1333032.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e11a6ea3425556d8565a54f35e586e31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a36a9dc09e4ed89c47993141ea124fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/129f1d6567bef09b00b7f37894e6dba7.png)
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名校
【推荐2】如图①,在矩形ABCD中,已知BC=8cm,点G为BC边上一点,满足BG=AB=6cm,动点E以1cm/s的速度沿线段BG从点B移动到点G,连接AE,作EF⊥AE,交线段CD于点F.设点E移动的时间为t(s),CF的长度为y(cm),y与t的函数关系如图②所示.
![](https://img.xkw.com/dksih/QBM/2021/12/30/2883259841789952/2883798950092800/STEM/d3aec2e5a0bd4749a43286f648a33d77.png?resizew=337)
(1)图①中,CG=______cm,图②中,m=______;
(2)点F能否为线段CD的中点?若可能,求出此时t的值,若不可能,请说明理由;
(3)在图①中,连接AF,AG,设AG与EF交于点H,若AG平分△AEF的面积,求此时t的值.
![](https://img.xkw.com/dksih/QBM/2021/12/30/2883259841789952/2883798950092800/STEM/d3aec2e5a0bd4749a43286f648a33d77.png?resizew=337)
(1)图①中,CG=______cm,图②中,m=______;
(2)点F能否为线段CD的中点?若可能,求出此时t的值,若不可能,请说明理由;
(3)在图①中,连接AF,AG,设AG与EF交于点H,若AG平分△AEF的面积,求此时t的值.
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【推荐1】如图1,在四边形
中,
,点
在边
上,且
,
.作
,交线段
于点
,连结
.
![](https://img.xkw.com/dksih/QBM/2022/9/23/3072676992532480/3123685719326720/STEM/eb4a67ef1193472c9843ae885694aaf6.png?resizew=544)
(1)求证:
;
(2)如图2,若
,
,
,求
的长;
(3)如图3,若
的延长线经过
的中点
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f90318c1a8d868d69fba0e8ab960bfc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3af4a9a466a3b1a794f6190c860806de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c0aa4c793921fdaa5d430cb90d78ca6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e15d9cb7e29df6d909524f653782a5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://img.xkw.com/dksih/QBM/2022/9/23/3072676992532480/3123685719326720/STEM/eb4a67ef1193472c9843ae885694aaf6.png?resizew=544)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c280fc03fdbcf37491a8573850a1df95.png)
(2)如图2,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d649afbddd907f0dfec1420f02f82fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b515965c22d2950b592c096c6e3bdfd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9392842583ac2a72a9385a7ee6886096.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
(3)如图3,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.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/8915a3686bd9b0e04223907bf56e89ba.png)
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解题方法
【推荐2】(1)在正方形
中,G是
边上的一个动点(不与
重合,以
为边在正方形
外作一个正方形
,连接
,如图1.直接写出线段
的关系________;
(2)将图1中的正方形
绕点C按逆时针方向旋转任意角度
,得到图2的情形,试判断(1)中的结论是否仍然成立?并证明你的判断;
(3)将上面题中的正方形都改为矩形,得到如图3、图4,若
,
试判断(1)中的结论是否仍然成立?请以图4为例,简要说明理由.
(4)在图4中,连接
,若
,直接写出
的值_________.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf80148409afb32ced0b4f59f1ba709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b1f13007c6d134c50004c62dc240707.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/514f68ab456b7442a050868ff65f6f71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/514f68ab456b7442a050868ff65f6f71.png)
(2)将图1中的正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b1f13007c6d134c50004c62dc240707.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
(3)将上面题中的正方形都改为矩形,得到如图3、图4,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44d534dfb4fa400fc52ecfd9f238681e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9d4a507467f22a1cb78a1ed1e7c012.png)
(4)在图4中,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/456c468f95e63bc67b9e03e1da1d47b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dca6c93d04aeaa0b77f24a1810e7b4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0f8db56e58f0f8fc881dfa737e3a0c4.png)
![](https://img.xkw.com/dksih/QBM/2021/5/2/2712451948355584/2726452690239488/STEM/78529abb-fd8c-458e-9d78-82a2e8c25966.png)
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