为培养同学们的数学思维,刘老师提出一个问题情境,供兴趣小组的同学们研讨,
问题情境:如图1,矩形
中,将
沿对角线
折叠,得到
,点A 的对应点为点M,
交
于点 N,连接
.
请仔细阅读以上研讨过程,完成下述任务.
(1)小明的理由中①的依据是 ;②的依据是 .
(2)如图2:
①小亮的结论是否正确? (不需证明)
②在
中,
与
满足什么数量关系时,可以得到
?请说明理由.
(3)如图3,平行四边形
中,
,按问题情境操作,当
是直角三角形时,请直接写出
的长.
问题情境:如图1,矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b9f0296c53918018745f4e3906e2dd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c884b508394b3ab50734b584d9ec783c.png)
刘老师:在这个问题情境下,你能得到什么结论? 小明:我能得到 ![]() 理由:如图1,∵ 矩形 ![]() ![]() ∴ ![]() ∵ ![]() ∴ ![]() ∴ ![]() ∴ ![]() ∵ ![]() ∴ ![]() ∴ ![]() 即 ![]() ∴ ![]() ∵ ![]() ∴ ![]() ∴ ![]() 小亮:不只是矩形有这样的关系,我们可以从“特殊到一般”,把矩形换为平行四边形.如图2,在平行四边形 ![]() ![]() 小红:小亮的思路很好,我发现如果在图2的 ![]() ![]() ![]() ![]() |
(1)小明的理由中①的依据是 ;②的依据是 .
(2)如图2:
①小亮的结论是否正确? (不需证明)
②在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3818a2c9919d358b4c3713396093822b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a9fe609f7851c20ad9a6892f080a0ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77fa0984a133ccad9c6e218d84b63a55.png)
(3)如图3,平行四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274a9d424e5c6777f6f6544bd3d66b9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cce3ac38fff90d90bce109cca298fc93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
更新时间:2024-04-12 11:32:37
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相似题推荐
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【推荐2】如图1,在平面直角坐标系中,A(a,0),B(b,0), C(-1,2),且
+
=0
(1)求a,b的值.
(2)在坐标轴上是否存在一点M,使
COM的面积=![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
ABC的面积,求出点M的坐标.
(3)如图2,过点C作CD⊥y轴交y轴于点D,点P为线段CD延长线上的一动点,连接OP,OE平分∠AOP,OF⊥OE.当点P运动时,
的值是否会改变?若不变,求其值;若改变,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58a71899a2574f5ea2f3e714fb61d708.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d46cfff060a5116f0a98fdd7d4dc7bf9.png)
(1)求a,b的值.
(2)在坐标轴上是否存在一点M,使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4cba95fc7d4853a243f8e3fb20ce70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4cba95fc7d4853a243f8e3fb20ce70.png)
(3)如图2,过点C作CD⊥y轴交y轴于点D,点P为线段CD延长线上的一动点,连接OP,OE平分∠AOP,OF⊥OE.当点P运动时,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69f2a7ebf0bb96fbe9fc9dabb63a8c12.png)
![](https://img.xkw.com/dksih/QBM/2021/4/21/2704469560541184/2705848049762304/STEM/76887ee7-242f-4398-8797-6d02fc401d90.png)
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【推荐1】如图矩形(长方形)
沿
折叠,使点D落在点E的位置,
与
相交于点F,若
,求
的长.
![](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/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d83e40b26f6a76e7897353f5699db4a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
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【推荐2】如图,在一张长方形纸片ABCD中,点E,F分别是CD和AB的中点.将这张纸片按图示方式折叠后,D,G,H三点共线,且G是DH中点.
![](https://img.xkw.com/dksih/QBM/2021/3/6/2671902911070208/2674399047696384/STEM/5c377bad61a14acda7de52499ee48b3b.png?resizew=119)
(1)∠DAH=__________;
(2)若AB=20cm,EG=__________cm.
![](https://img.xkw.com/dksih/QBM/2021/3/6/2671902911070208/2674399047696384/STEM/5c377bad61a14acda7de52499ee48b3b.png?resizew=119)
(1)∠DAH=__________;
(2)若AB=20cm,EG=__________cm.
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名校
【推荐1】有一斜坡
,其坡度为
,顶部
处的高
为
在同一水平地面上.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/6/f6c3f086-1289-4a30-ae7a-23fa2b72c8e7.png?resizew=193)
(1)求斜坡
的水平宽度
的长;
(2)矩形
为长方体货柜的侧面图,其中
.将该货柜沿斜坡向上运送,当
,求点
离地面的高.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09824eebae0979a2b6b1fcc4d04bdee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ea3499cd5e31207e4d980127e93955f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/6/f6c3f086-1289-4a30-ae7a-23fa2b72c8e7.png?resizew=193)
(1)求斜坡
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(2)矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2257da1e2425f2ea9ac7440f52659ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee465313f6f929f1a405638293cfded8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de7cc8fc424f249c6a6d97762a617dbd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
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【推荐2】如图,在△ABC中,D,E分别是AC,AB上的点,∠ADE=∠B.△ABC的角平分线AF交DE于点G,交BC于点F.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/19/3ae63f8b-5830-4318-bcdd-f8a2f15b4dc8.png?resizew=124)
(1)求证:△ADG∽△ABF;
(2)若
,AF=6,求GF的长.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/19/3ae63f8b-5830-4318-bcdd-f8a2f15b4dc8.png?resizew=124)
(1)求证:△ADG∽△ABF;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f7b7aef85de2d4babbc025d4329007.png)
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【推荐3】对心曲柄滑块机构广泛应用于蒸汽机、内燃机、空压机以及各种冲压机器中.如图1是对心曲柄滑块机构的模型示意图,滑块B和曲柄
的O端在一条直线上,曲柄
绕回转中心O整周转动的过程中,通过连杆
使滑块B在直线
上往复运动.记直线
与
交于C,D两点(点D在点C的左侧).
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/10/29ffb999-ac89-4950-8a89-ca34eb974f58.png?resizew=156)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/9/fcb1bb88-c00c-4657-86a9-9d2e889f2f39.png?resizew=660)
(1)若曲柄
的长度为
,连杆
的长度为
,则滑块B到回转中心O的最小距离为
,最大距离为
.(用含a,b的式子表示)
(2)当连杆
与
相交于点E时,连接
,如图2所示.若
平分
,求证:
.
(3)当连杆
与
相切时,连接
,如图3所示.若曲柄
的长度为
,
,求连杆
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/10/29ffb999-ac89-4950-8a89-ca34eb974f58.png?resizew=156)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/9/fcb1bb88-c00c-4657-86a9-9d2e889f2f39.png?resizew=660)
(1)若曲柄
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18703d241f6fe0a0dedcc815603322fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f552d03a658afcdb420031762c79cae7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
(2)当连杆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f677175ac1313c715bf876123f7573c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c3d2cba96f6f03520c0b3f6e4da03e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4189a0821a0ffab9dc171ecd279ba442.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1225a8f430f7dad70b0c6173d3b6a99c.png)
(3)当连杆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbe9f7abf7bcf4e1aa2579cd191d7761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d985475d471e6ddc3a0ae5795750af5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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【推荐1】在
中,
,a,b,c分别为
,
,
的对边,根据下列条件求出直角三角形的其他元素(边长精确到0.01):
(1)
;
(2)
;
(3)
;
(4)
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f8f88798ec42a58dccd212586382b23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e3262fc038bbec5e7c8cc47df08bef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3818a2c9919d358b4c3713396093822b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/febc9a89d0d1c97b88c0f4acd32b4e67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/194741f4d2ae7ee44cafca780361446a.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fa26a19330e712731c7a1cd2c8cd6bb.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bec24bc93b3d469804774e244ee213d5.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f0ca3b8765e142e9c3c0065d9b9e97f.png)
(4)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a4d500cbeee4b3057821be4af4f5551.png)
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【推荐2】已知,在
中,
以
为直径的
与
交于点
将
沿射线
平移得到
连接![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bd97740e8f35c9c05f8bb53c0f1cb47.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/13/92546a1b-9475-4d60-bc18-11c0e6e369de.png?resizew=389)
(1)如图1,
与
相切于点
.
①求证:
;
②则
____.
(2)如图2,延长
与
交于点
将
沿
折叠,点
的对称点
恰好落在射线
上.
①求证:
;
②若
求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3262baa85fbc38a53855193395106fd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e47e91cdad2fa2f170c8add31d90676.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbc1d9204031c932719cb8159b77a5e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bd97740e8f35c9c05f8bb53c0f1cb47.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/13/92546a1b-9475-4d60-bc18-11c0e6e369de.png?resizew=389)
(1)如图1,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2cc98477f5f6950404402e00d0f1414.png)
②则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4ae83503b7a346bf502f81b09bff6bd.png)
(2)如图2,延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f32395bce0f32cf51671acddce256bc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d320e2347f8f06941d717766676595c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
的对称点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/480c4077a0e7e5cc4fbea3a2c772cfc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48ad8115387158c8dda205d26968ba11.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5b2613d29c97d6962cbe1472a84f44a.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a952f176a23f3aa69482bebbd15613b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
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