名校
解题方法
1 . 如图,在四棱锥P-ABCD中,底面ABCD为平行四边形,
,
,
,
,
,点N在棱PC上,平面
平面
.
;
(2)若
平面
,求三棱锥
的体积;
(3)若二面角
的平面角为
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e918b70b02a73685e3c536c7f380e2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced06b71073e1bb777f326f06016ce17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3ad66112b09c909cab417085702ec00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37002ada5d194d4d062fa3285d7d9824.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b235d0737ddc0d2c85abd4484c10d76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f04c222223dae9ef27d4c132534d9848.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e9e5f1cfea3643c30c21732073a11ef.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/373f735f0f04d11f1951eaef1bb78b6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfaf581b4f42a25087f7eee23a7d66b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5a94e7ab62cf6374d2e4c6d7240a271.png)
(3)若二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/957c57e018a7086ff1c1f823e02b7ef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15615de1a6df206dbd081251f676578e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50cafe199913787a939fe9e100924023.png)
您最近一年使用:0次
解题方法
2 . 在直三棱柱
中,
,点
是
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/23/ffbf31c6-3e48-4e9b-83e9-9ee867196f60.png?resizew=114)
(1)求异面直线
所成角的余弦值;
(2)求直线
与平面
所成角的正弦值;
(3)求直线
到平面
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5448308469294d862f2c761ace330aeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/23/ffbf31c6-3e48-4e9b-83e9-9ee867196f60.png?resizew=114)
(1)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/881a85d9088d781ba1bec7ab3e02de49.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b470c4e195cf7a07b7a331ce4b436e03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2ce0eeb7a6d6c7806cf2352b9fe15c2.png)
(3)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e26d9636ad77369535852c6e4493446a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2ce0eeb7a6d6c7806cf2352b9fe15c2.png)
您最近一年使用:0次
名校
解题方法
3 . 如图,在四棱锥
,
平面
,底面
是直角梯形,其中
,
,
,
,
为棱
上的点,且
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/14/e86845b6-fc97-4b95-a07d-635e5a66ae04.png?resizew=164)
(1)求证:
平面
;
(2)求平面
与平面
所成夹角的正弦值;
(3)求点
到平面
的距离.
![](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/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34e0a957a55460c72673c0f2ee90dbb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1134c8e3440abb6cd385af2c169037fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00bab2c27eac56fffa4cd7dbe1dcdf1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abfb2735e1683a6ae86b5b97a0032e4f.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/b41744ec71119e7264ef9673a35805a8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/14/e86845b6-fc97-4b95-a07d-635e5a66ae04.png?resizew=164)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8d2d217e9bcd059908f117dfc4d4259.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
(3)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
您最近一年使用:0次
解题方法
4 . 如图,
且
,
,
且
,
且
,
平面
,
,M为棱
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/13/ead5b860-4040-40db-af0f-f38a12e0c74b.png?resizew=149)
(1)求证:
平面
;
(2)求直线
与平面
所成角的正弦值;
(3)求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10df84d553a8826a7ce9bff4bf0d95b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64f1161e0345b3646c71365430dccbb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdb2dd10731b99c0f4f89ee957f8a239.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1989dc6aef61c294690d2105c72e894a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99188a6a00aabcd6936044139c771b1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/155de273b3d3857761ef315adb514b4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7cb1376856b53c9d7a721dd92564f84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cf187bc2ede965870b90757b495f53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6df76b0fddc037620e368d44cc30a791.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63e36329f5e0979f5ee776ac5d06327.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/13/ead5b860-4040-40db-af0f-f38a12e0c74b.png?resizew=149)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c8ccd4181f956f6e0140bf0ab8f0716.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0701f67727b0fc8100cfb5e20ec27d9b.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0701f67727b0fc8100cfb5e20ec27d9b.png)
(3)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43819ab7b268a6293a9251935b594690.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0701f67727b0fc8100cfb5e20ec27d9b.png)
您最近一年使用:0次
5 . 如图,在棱长为2的正方体
中,
为
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/13/34441aed-3e4d-49d6-8091-33ef0c441207.png?resizew=167)
(1)求证:
平面
;
(2)求直线
与平面
所成角的正弦值;
(3)求平面
和平面
的夹角的余弦值.
![](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/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/13/34441aed-3e4d-49d6-8091-33ef0c441207.png?resizew=167)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/debdc6632a4877e5131d3da25cda8b89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea55a7e39361987096953d3a3ee1eaa4.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83c09eec4e14a861af83d7828797d176.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea55a7e39361987096953d3a3ee1eaa4.png)
(3)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2331bccb6ebf5b9fd639df994f575a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea55a7e39361987096953d3a3ee1eaa4.png)
您最近一年使用:0次
解题方法
6 . 如图,已知
平面
,
,
,
,
,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/11/0a5d7a9b-1d84-43bf-8ffa-fadd382d50dd.png?resizew=159)
(1)求证:
平面
;
(2)求平面
与平面
的夹角的余弦值;
(3)求点
到直线
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/000a5d60075d7f1b9471cb12c18ebecc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4adf90a8c2b29334cdc5aa5b554991f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdb2dd10731b99c0f4f89ee957f8a239.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95ac28e265e944e323bed24e334969d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d2220e68f9d97641a8074b1f7fe0a33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10b66db4a8fbc6b0dc3dc160246b3533.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/deb3a2b9ab304181ee5ff73b97a196ec.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/11/0a5d7a9b-1d84-43bf-8ffa-fadd382d50dd.png?resizew=159)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72a6ca1d6b37072bd7df0cf77737bb79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9539f8fb13345b449274b67bbda995db.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e22ebcc4aa98d46366df48f751a5f368.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9539f8fb13345b449274b67bbda995db.png)
(3)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e26d9636ad77369535852c6e4493446a.png)
您最近一年使用:0次
解题方法
7 . 如图,正方形
与梯形
所在平面互相垂直,已知.
//
,
,
点P为线段EC的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/10/8b134d4e-7962-42ff-9313-142637538d58.png?resizew=157)
(1)求证:
∥平面CDE;
(2)求直线DP与平面
所成角的正弦值;
(3)求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ecc1cb55a57dde481f8dd07ab150676.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdb2dd10731b99c0f4f89ee957f8a239.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/987e2ad8478919f12a8cd0d7dd3309e5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/10/8b134d4e-7962-42ff-9313-142637538d58.png?resizew=157)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
(2)求直线DP与平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ae8768996ca9a0f2c5d9a19abbd54df.png)
(3)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ae8768996ca9a0f2c5d9a19abbd54df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10fc7991ea17d54ff5f4445ac5699463.png)
您最近一年使用:0次
解题方法
8 . 如图,在三棱柱
中,
平面ABC,
,
,D、M是线段BC、
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/8/403b46ba-9994-45bc-837c-57402b89f54e.png?resizew=161)
(1)求证:
平面
;
(2)求点
到平面BCM的距离;
(3)求直线
与平面BCM所成角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5845ccc0d735dc14c92a8926d9b1def6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c4559d27e3905980d1a4f1856f07de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2af2626608f61a4cfbb86494bd6df0e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/8/403b46ba-9994-45bc-837c-57402b89f54e.png?resizew=161)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/554923047631d16320c2ba39abeee99c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efb87785b2842459c59b2571aac7374b.png)
(2)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
(3)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b470c4e195cf7a07b7a331ce4b436e03.png)
您最近一年使用:0次
9 . 如图,在四棱锥
中,底面
为矩形,侧棱
底面
,
,
是
的中点,点
在棱
上且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6e638e80f5e7a6379193c1ac6a8a98.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/6/0cf4d636-b735-4c5f-8ed7-092fca8828f7.png?resizew=145)
(1)求证:
平面
;
(2)求平面
与平面
夹角的余弦值;
(3)求点
到平面
的距离.
![](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/5a1b49f64e0065edad868b25e9fcada3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb886ce5fa68b1bafeed307589576348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6e638e80f5e7a6379193c1ac6a8a98.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/6/0cf4d636-b735-4c5f-8ed7-092fca8828f7.png?resizew=145)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/373f735f0f04d11f1951eaef1bb78b6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/466fabcaac59132fea648ff35342ec9d.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/466fabcaac59132fea648ff35342ec9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/218054144a13435580cd132b9459546c.png)
(3)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/466fabcaac59132fea648ff35342ec9d.png)
您最近一年使用:0次
名校
10 . 四棱锥P-ABCD中,
平面ABCD,
,
,
,
,E是
的中点,点F在线段
上,且满足
.
(1)求证:
平面
;
(2)求平面
与平面
夹角的余弦值;
(3)在线段
上是否存在点Q,使得
与平面
所成角的余弦值是
,若存在,求出
的长;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1b49f64e0065edad868b25e9fcada3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee8ef58be8708144272538ee427fb92c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1134c8e3440abb6cd385af2c169037fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5c2f9f2c5e1def9d1e4c051e5143054.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa6e011c6fc0940270619a401d6f6cbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fbae68020a497f0c021bea162bcebaf.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/4/9a224c01-c909-4bdd-8710-9e7264ccdee9.png?resizew=120)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7440b41636c761b0910639e310ff7dfb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f3cac89b6817ad3603506236ea8e501.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeb5255e2159617505e0c87d01437a57.png)
(3)在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632917e61f4208959686d118c7f19231.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee72fd8a5a52d08a4fddcf0830a8e103.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1174142f3bba761585b6bc2653009b36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d454c82d9e52747563d47b68099249.png)
您最近一年使用:0次