“风筝”是中国传统文化中不可或缺的一部分,距今已有2000多年的历史.相传在东周春秋时期,墨翟以木头制成木鸟,是人类最早的风筝起源.后来鲁班用竹子,改进墨翟的风筝材质,直至东汉期间,蔡伦改进造纸术后,坊间才开始以纸做风筝,称为“纸鸢”.到南北朝时,风筝开始成为传递信息的工具;从隋唐开始,由于造纸业的发达,民间开始用纸来裱糊风筝;到了宋代的时候,放风筝成为人们喜爱的户外活动.风筝主要由骨架、风筝面、尾翼、提线、放飞线五部分组成.如图(1)就是一个由菱形的风筝面ABCD和两个直角三角形尾翼
和
所组成的风筝.其中
,
,
,
,
.现将此风筝的两个尾翼分别沿
折起,使得点P与点Q重合于点S,并连结
,得到如图(2)所示的四棱锥
.
平面
;
(2)若E为棱
上一点,记![](https://staticzujuan.xkw.com/quesimg/Upload/formula/968088978992ba767468dec2dc436350.png)
①若
求直线
与平面
所成角的正切值;
②是否存在点E使得直线
与直线
所成角为
,若存在请求出
的值,若不存在请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d41984f53bb280ba8b5ac00a52ce2825.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d96645a3530e72d5d733d2c72147d340.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb8fb552b9e21dbaba74d11aa747790.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37002ada5d194d4d062fa3285d7d9824.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d899b31ac8800258c52e86a70e7ab9ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4aca5534bce25acaeb7379deed8f8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a0f2f623bbe6beb1fdbc767bc1ba70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a887678ca42faa3d289e2b6460790b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b5e290c6b2c5508a3bf6117afbf7e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e56fdf217165748fafe938b64fa08179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5000fea066102e62cf2128ccbbd2b3e3.png)
(2)若E为棱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a6e2867f32d3f1c3cd36cd3a11a8580.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/968088978992ba767468dec2dc436350.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1c3ea872a20fdc1843cb5ffce8a554.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5000fea066102e62cf2128ccbbd2b3e3.png)
②是否存在点E使得直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
更新时间:2024-05-24 14:35:50
|
相似题推荐
解答题-证明题
|
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(0.65)
解题方法
【推荐1】如图,棱长为2的正方体
中,点
,
,
分别是棱
,
,
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/2/6ed25543-273f-46a9-97a7-9b00354c7661.png?resizew=148)
(1)求证:直线
平面
;
(2)求异面直线
和
所成角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/394c5d2f55221975503be8aa18022480.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/2/6ed25543-273f-46a9-97a7-9b00354c7661.png?resizew=148)
(1)求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63a779876cdfb2c489ad0eaed0f73e6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/152cfd5011c94e02c1a9cbbd4d8f58bb.png)
(2)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
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【推荐2】在直三棱柱
中,
,D是AB中点,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/2/3e35757b-f903-4baf-b5c0-98b29bd7ab81.png?resizew=166)
(1)求异面直线
与
所成角的大小;
(2)证明:
平面
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b51da47ab8433342f7a319e412fefae.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/2/3e35757b-f903-4baf-b5c0-98b29bd7ab81.png?resizew=166)
(1)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7f6f93171329d508d491143b9d71f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89006cac018a9875f65ed7bd429c61bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0772e18d6605a5c10ade8b5ecd1ae77.png)
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【推荐3】如图,在四棱锥P-ABCD中,底面ABCD是矩形,
PA⊥底面ABCD,E是PC的中点.已知AB=2,
AD=2
,PA=2.求:
(1)三角形PCD的面积;
(2)异面直线BC与AE所成的角的大小.
PA⊥底面ABCD,E是PC的中点.已知AB=2,
AD=2
![](https://img.xkw.com/dksih/QBM/2012/6/15/1570887624867840/1570887630348288/STEM/e150781c0cf24d2aa2e83584f0e2e7c8.png?resizew=21)
(1)三角形PCD的面积;
(2)异面直线BC与AE所成的角的大小.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/21/bd392f6c-1072-49b0-ab34-f8a9301293f6.png?resizew=153)
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【推荐1】如图,在四棱锥
中,底面
为正方形,侧面
是正三角形,侧面
底面
,E是
的中点.
(1)过点E在面
内画一条直线l,使得
,写出做法,并说明理由;
(2)设直线l与
交于F点,求
与底面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852aabd89edffc1b94344ff3f1f31ccd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/25/e556d65b-b8e6-43c7-a5d1-fb241d928f56.png?resizew=170)
(1)过点E在面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e091f5bbf66bd625cb1adab21d8c75b.png)
(2)设直线l与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/735056c174e8dd7906257a2a50a962a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
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【推荐2】如图,三棱柱ABC﹣A'B'C',AC=2,BC=4,∠ACB=120°,∠ACC'=90°,且平面AB'C⊥平面ABC,二面角A'﹣AC﹣B'为30°,E、F分别为A'C、B'C'的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/24/e2624112-5175-4737-b103-aeb470da0aff.png?resizew=250)
(1)求证:EF∥平面AB'C;
(2)求B'到平面ABC的距离;
(3)求二面角A﹣BB'﹣C'的余弦值.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/24/e2624112-5175-4737-b103-aeb470da0aff.png?resizew=250)
(1)求证:EF∥平面AB'C;
(2)求B'到平面ABC的距离;
(3)求二面角A﹣BB'﹣C'的余弦值.
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【推荐3】在四棱锥P-ABCD中,底面ABCD为直角梯形,
,
.
,
,PD⊥DC.
![](https://img.xkw.com/dksih/QBM/2022/12/16/3132150951485440/3133040264372224/STEM/11f5df0210af4369b1836b081d4e9ef2.png?resizew=276)
(1)证明:
面ABCD;
(2)求直线BD与平面PBC所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10df84d553a8826a7ce9bff4bf0d95b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaa2efb8f9721bc15d5243e878346fba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fe302adc32c71bb90a8f007501ff132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f83a04565a8ebaa111894b724b0ba266.png)
![](https://img.xkw.com/dksih/QBM/2022/12/16/3132150951485440/3133040264372224/STEM/11f5df0210af4369b1836b081d4e9ef2.png?resizew=276)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
(2)求直线BD与平面PBC所成角的正弦值.
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【推荐1】如图,在四棱锥
中,
,
,
,E为棱
的中点,
平面
.
平面
;
(2)求证:平面
平面
;
(3)若二面角
的大小为
,求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef699f5dc072b853cfe700c6f1abbbae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a0e5697eca3f5205cb7b343648240bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/639bec6242a4b3f7bfb4b7033a67328c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307807ee10071bafbe922eb18d2517d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9068f29d671d76d1e95ba3a4eaff5b96.png)
(2)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4aa9084b8fe0fe05c4388d1f835587b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f571a1aac46c6d0cf440c0ec2846bf9.png)
(3)若二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1636b4530c0b42d0e0b649e90e3b9e85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29e0c2455a9e796bba6861503f0fe31b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f571a1aac46c6d0cf440c0ec2846bf9.png)
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【推荐2】如图,四棱锥
中,
底面ABCD,
是矩形,
是棱
的中点,
,
.
![](https://img.xkw.com/dksih/QBM/2015/1/14/1571960773165056/1571960779128832/STEM/77b945af6ccb40b580dd81bceb1ae313.png?resizew=163)
(1)证明:
平面
;
(2)求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c2753753faf2cb9a0003aa8e3945159.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://img.xkw.com/dksih/QBM/2015/1/14/1571960773165056/1571960779128832/STEM/77b945af6ccb40b580dd81bceb1ae313.png?resizew=163)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa69a2247ad4d5231aa361349b12f97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c66d99a6a8415ddad22bbed33b64cfb.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
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【推荐1】如图,在四棱锥
中,底面ABCD是矩形,平面
平面SBC,
,M是BC的中点,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/a8eb0373-01ac-401f-aae7-181deb95a4db.png?resizew=161)
(1)求证:
.
(2)若
,求四棱锥
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf9a6db3571fa57bfa2d5e4d44c51b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4cab41e3c3e1b04f0cff21aca315238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced06b71073e1bb777f326f06016ce17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef0402dd5ae3db10281f9f1e11738bcb.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/a8eb0373-01ac-401f-aae7-181deb95a4db.png?resizew=161)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2919a92ec8fdae2a7b8511fff31fa65.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54ecf4712f4ce3a52ccc695c1774cbe5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
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解题方法
【推荐2】如图,在三棱锥
中,
,点D是棱
的中点.
![](https://img.xkw.com/dksih/QBM/2022/1/22/2899907775348736/2921167411814400/STEM/fd1e8f0d-a3a6-446e-9983-6ab9d71717d5.png?resizew=168)
(1)求证
;
(2)若
,点E在棱
上,且
,求点C到平面
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee5a6d7e06b468a8921835d10c44a827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://img.xkw.com/dksih/QBM/2022/1/22/2899907775348736/2921167411814400/STEM/fd1e8f0d-a3a6-446e-9983-6ab9d71717d5.png?resizew=168)
(1)求证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c65334978b0519b379910dfc4acf8344.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f60c68102af27fbf7a443be683e733d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e73c8c1d2ba6b29b301380a45dfbcdd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b37793a3a810e823e10c340986f55ddd.png)
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