探索并解决下列问题:
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/13/37482619-904f-40ba-bb9b-056e274b0315.png?resizew=362)
(1)如图1,长方形
的边
,
,点
从点
出发,沿
的路径以每秒
的速度运动,到达点
时停止运动
设运动时间为
.
①用含
的代数式表示三角形
的面积;
②当三角形
的面积为
时,求
的值.
(2)如图2,已知长方形
,以它的对角线
为边作另一个长方形
,其中
经过点
现有一点
在长方形
内随意运动,连接
和
若三角形
的面积为
,
,那么随着点
的运动,封闭图形
的周长是否有最小值?如果有,请求出这个最小值;如果没有,请说明理由.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/13/37482619-904f-40ba-bb9b-056e274b0315.png?resizew=362)
(1)如图1,长方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6747441b260ca043446b5d472fece440.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9547f6b5c7c54d216ecea72becb7d373.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c27c429dc990ea4183f060f5d390a36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c78d0ab561d0c9bb9099772c596af8bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54e1d0f65817ba32a732040518f41440.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d744165d6aad705f64de41a9d931750a.png)
①用含
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745e0525a41fe2e2a7739c75a942290b.png)
②当三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/745e0525a41fe2e2a7739c75a942290b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8c4c029e552954bd493b49aeab82d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(2)如图2,已知长方形
![](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/1507da3daed983c2f355d4caebb66d72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/789cac70cd4769dcdcf49f649d5cb5ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c401d2901addce8ab172c03326f80529.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4eb7e9ad5486cf1c5e506b20c5469e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f2c7aaa24c251b10f1c7a0dce744602.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a4fdeb084a614770082db186f8ca4f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a418ec1bbca805933d170b2adaffa97.png)
更新时间:2024-01-12 23:12:42
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相似题推荐
解答题-问答题
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名校
【推荐1】如图,O是直线PQ上一点,OM是直线PQ上方过点O的一条射线,
.若射线OA在∠MOQ内,∠AOM的大小为
.射线OB在直线PQ上方,且
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/25/0b9ed92b-5aca-4eb9-a41f-390388716518.png?resizew=288)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/25/09ddb55c-4817-4e53-bff6-fd57ed971f6b.png?resizew=273)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/25/77412a9f-2882-444f-b2b2-62ab36e494c7.png?resizew=275)
(1)用t的代数式表示
________;
(2)当
时,求t的值;
(3)若射线OD在
内,且
,当OA,OB,OD三条射线中的一条射线是另两条射线组成的夹角的平分线时,请求出t的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69af676d3c92e7dd4efd77fcd09e30c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42dc70a05db0ef23fc37aefc2e33d64b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b090804dba52f520ddb928b2d3f75181.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/25/0b9ed92b-5aca-4eb9-a41f-390388716518.png?resizew=288)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/25/09ddb55c-4817-4e53-bff6-fd57ed971f6b.png?resizew=273)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/25/77412a9f-2882-444f-b2b2-62ab36e494c7.png?resizew=275)
(1)用t的代数式表示
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d18eaed610d6c54cee69978380f6853f.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38c211a24e544ded208ea0a2a2aa59bf.png)
(3)若射线OD在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c3a3bf63440a2fdd769162516fb0e1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf5beaa08ed4c7d9d1c39c526e5ff135.png)
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解答题-计算题
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【推荐2】任何一个正整数n都可以这样分解:
(p、q是正整数,且
),则n的所有这种分解中,如果两因数p,q之差的绝对值最小,我们就称
是n的最佳分解,并规定:
.
例如:18可以分解成
或
,则
.
(1)计算:
、
.
(2)如果一个三位正整数
(
,x,y为自然数),交换其个位上的数与百位上的数得到的新三位正整数加上原来的三位正整数所得的和恰好能被11整除,那么我们称这个数t为“心意数”.
①求所有满足条件的“心意数”t;
②对于满足“心意数”t中的x,y,设
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/171686ae7ae5b7294230d2499a202ea1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627e7a746dea05435fbbae22d8d8d2ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e35f5dbb594bc6c1f7480715a7cdd3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17584821598ab07d97a1e8a5fb6e6dd9.png)
例如:18可以分解成
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/153913d031e5b7a703047816a4fb9454.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89433d60e02fb414435c68aeb984f689.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ff8bc9b1cc841a272a2b44f91f76a78.png)
(1)计算:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0207083221da502f0eb1fdef56ecc8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b35b21249d2b512193fa6e6b8db30885.png)
(2)如果一个三位正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c643f654cce2eeb5361c440ebb4d5557.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d6fd65750c0c4b276bc7aa9e16faef7.png)
①求所有满足条件的“心意数”t;
②对于满足“心意数”t中的x,y,设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/137ab253f9ea554258c43b4a20af1cf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bde5899ab8503b1da059bf37311bd819.png)
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【推荐1】一副三角尺按如图所示摆放,将三角尺
绕点
以每秒15°的速度顺时针旋转,同时,三角尺
也绕点
以每秒
的速度逆时针旋转,三角尺
的直角边
与
重合时停止旋转,三角尺
也停止旋转,设运动的时间为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/4/5/3fe4b503-c9f6-43aa-9d0d-d960c23368c8.png?resizew=229)
(1)用含
的代数式表示:
______;
______;
(2)当
为何值时,
?
(3)下列三个问题,依次按易、中、难排列,对应的满分值为2分、3分、4分,请根据自己的认知水平,选择其中一个问题完成做答.
①当
为何值时,
与
重合?
②当
为何值时,
?
③在旋转过程中是否存在这样的
,使得射线
是由射线
、射线
、射线
中的其中两条组成的角的平分线?如果存在,请求出
的值;如果不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a4d781525777c7b5284dffc70b2a28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/741a44528782580d24249e8075b8cd9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a4d781525777c7b5284dffc70b2a28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c8ffe24cf9f327aeb241225ab15ab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/4/5/3fe4b503-c9f6-43aa-9d0d-d960c23368c8.png?resizew=229)
(1)用含
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa4bb72b653a509e6a61ecef9e83dd92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89a33e3a2df3099c888511c17812060b.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4161b824d724063e21686ef5fc901c5e.png)
(3)下列三个问题,依次按易、中、难排列,对应的满分值为2分、3分、4分,请根据自己的认知水平,选择其中一个问题完成做答.
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
②当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f99f724bb3614b40d63b284abaa7993f.png)
③在旋转过程中是否存在这样的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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【推荐2】如图1,在四边形ABCD中,
,∠B=90°,AD=acm,BC=bcm,并且a,b满足b=
+8,若动点P从A点出发,以每秒0.5cm的速度沿线段AD向点D运动;点Q从C点出发以每秒2cm的速度沿CB方向运动,当P点到达D点时,动点P、Q同时停止运动,回答下列问题:
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/9/baf6d49d-4849-4dd0-9700-62a0d7470b81.png?resizew=481)
(1)AD=______cm,BC=______cm.
(2)设点P、Q同时出发,并运动了x秒,求当x为多少秒时,四边形PQBA是矩形.
(3)如图2,若四边形ABCD变为平行四边形ABCD,AD=BC=6cm,动点P从A点出发,以每秒0.5cm的速度沿线段AD向点D运动;动点Q从C点出发以每秒2cm的速度在BC间往返运动,当P点到达D点时,动点P、Q同时停止运动,设P、Q两点同时出发,并运动了t秒,求当t为多少秒时,以P,D,Q,B四点组成的四边形是平行四边形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4adf90a8c2b29334cdc5aa5b554991f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db298b1b63d4de4b487616b1c6c600bc.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/9/baf6d49d-4849-4dd0-9700-62a0d7470b81.png?resizew=481)
(1)AD=______cm,BC=______cm.
(2)设点P、Q同时出发,并运动了x秒,求当x为多少秒时,四边形PQBA是矩形.
(3)如图2,若四边形ABCD变为平行四边形ABCD,AD=BC=6cm,动点P从A点出发,以每秒0.5cm的速度沿线段AD向点D运动;动点Q从C点出发以每秒2cm的速度在BC间往返运动,当P点到达D点时,动点P、Q同时停止运动,设P、Q两点同时出发,并运动了t秒,求当t为多少秒时,以P,D,Q,B四点组成的四边形是平行四边形.
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【推荐1】(1)问题再现:学习二次根式时,老师给同学们提出了一个求代数式最小值的问题,如,求代数式
的最小值;小强同学发现
可看作两直角边分别为
和2的直角三角形斜边长,
可看作两直角边分别是
和3的直角三角形的斜边长.于是构造出下图,将问题转化为求
的斜边
和
的斜边
的和的最小值,易得
、
、
三点共线时,
取得最小值,即线段
的长,进而求得
的最小值是_________.
已知
,
均为正数,且
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bc0a2547bab315f92ce3027087ffeb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90216f30736912b12d5ad16bb4264a28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03cbc03d82110fcc1dbdad6d4a0ba692.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf6dddd3712d5992eeac94d726a45f97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10ce5d6ca24e249f7172e51c154509b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86ba4917a177be27018540187a1cfbca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab15f71acf24a48b4ea4efac5f09b718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bc0a2547bab315f92ce3027087ffeb8.png)
已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12b40b1544e62be8b9e9f4dc9f2c0c74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d4bae74735a08ffd73597feddf004bb.png)
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解答题-证明题
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【推荐2】如图1,
和
都是等边三角形,连接
,将
绕点B逆时针
得到
,连接
,
.
(1)连接
,求证:
;
(2)将图1中的
绕点A顺时针旋转
,如图2,当
,且
时,求证:四边形
为菱形;
(3)如图3,连接
,取
,
的中点M,N,若
,
,将
绕点A顺时针旋转α,当
,线段
取最小值时,直接写出线段
的长度.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/1/95fb03a0-e8ac-4609-8295-edcddc08ba56.png?resizew=501)
(1)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/708f18bf26949797ad5c3d23c0783317.png)
(2)将图1中的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f64f0db3ac82ba45e8272f63dcc65a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7059ac1fdadf0bc6474d78c03ac7e121.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b312dab930cbbb9a4bb1a99f044dab73.png)
(3)如图3,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305a88d4e0249bd16d48eda01331d2d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d27bd71d79cb19eb554175e4ef0867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6211f52496832c3bc5f07e9d4a9bba8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
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