(本题满分12分)如图,底面为菱形的四棱锥P-ABCD中,∠ABC=60°,AC="1," PA="2," PB=PD=
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(本题满分12分)如图,底面为菱形的四棱锥P-ABCD中,∠ABC=60°,AC="1," PA="2," PB=PD= ![]() ![]() (Ⅰ)证明:PA⊥平面ABCD; (Ⅱ)若AN为PD边的高线,求二面角M-AC-N的余弦值. |
![(本题满分12分)如图,底面为菱形的四棱锥P-ABCD中,∠ABC=60°,AC=](/uploads/image/z/6045847-7-7.jpg?t=%EF%BC%88%E6%9C%AC%E9%A2%98%E6%BB%A1%E5%88%8612%E5%88%86%EF%BC%89%E5%A6%82%E5%9B%BE%EF%BC%8C%E5%BA%95%E9%9D%A2%E4%B8%BA%E8%8F%B1%E5%BD%A2%E7%9A%84%E5%9B%9B%E6%A3%B1%E9%94%A5P-ABCD%E4%B8%AD%EF%BC%8C%E2%88%A0ABC%3D60%C2%B0%EF%BC%8CAC%3D%221%2C%22+PA%3D%222%2C%22+PB%3DPD%3D)
证明:见解析;
(Ⅱ)
![](http://img.wesiedu.com/upload/b/4f/b4f5814f46ad39beba3af312ba6580d4.jpg)
本试题主要是考查了线面垂直的判定和二面角平面角的求解的综合运用。
(1)要证明线面垂直,要通过判定定理线线垂直得到线面垂直,关键是证明
![](http://img.wesiedu.com/upload/f/a5/fa5e8129baaaccb9689ffcdba5461932.jpg)
![](http://img.wesiedu.com/upload/7/45/745ef9af0bca70286e34debb50553036.jpg)
(2)建立空间直角坐标系,然后表示出平面的法向量与法向量的夹角,进而求解二面角的平面角的大小的求解。
![](http://img.wesiedu.com/upload/e/93/e93f4f9e78c73300f03bca63310680cc.jpg)
证明:(Ⅰ)∵菱形ABCD中∠ABC=60°,
∴
![](http://img.wesiedu.com/upload/c/8e/c8e6b75e5d58760117520993e1ea33bb.jpg)
∴
![](http://img.wesiedu.com/upload/3/21/32192b7b368d55aca5a60e62adeb3415.jpg)
又∵
![](http://img.wesiedu.com/upload/3/23/323db58549279ad72a247a6c5b49f8fc.jpg)
![](http://img.wesiedu.com/upload/c/1e/c1ec7430647bc16d43eae6a91ba86126.jpg)
∴有
![](http://img.wesiedu.com/upload/b/46/b467394534ba03659914732cd7cd6249.jpg)
![](http://img.wesiedu.com/upload/d/de/dde0cb988ed5e7e9b3817efa7c5aee56.jpg)
∴
![](http://img.wesiedu.com/upload/6/b3/6b3e0fd0638ff80f03ddfdcb7ef1afc9.jpg)
![](http://img.wesiedu.com/upload/b/f4/bf4f0f19ec6b65ff709ba031eeb12712.jpg)
∴
![](http://img.wesiedu.com/upload/f/a5/fa5e8129baaaccb9689ffcdba5461932.jpg)
![](http://img.wesiedu.com/upload/7/45/745ef9af0bca70286e34debb50553036.jpg)
![](http://img.wesiedu.com/upload/9/8e/98e3991c16699447cdb7e33fe31dfe06.jpg)
∴
![](http://img.wesiedu.com/upload/2/56/25616d30f45fd119e25f447bafcd3ad7.jpg)
![](http://img.wesiedu.com/upload/e/02/e029a1166c481bd1712c43c3ff25fe10.jpg)
(Ⅱ)取BC中点E,连结AE,则AE⊥BC.以点A为坐标原点,AE为x轴正向,AD为y轴正向,AP为z轴正向建立空间直角坐标系,则
![](http://img.wesiedu.com/upload/c/cd/ccd0ef6657115623f1f0d97b5ff13530.jpg)
在
![](http://img.wesiedu.com/upload/c/8e/c8e6b75e5d58760117520993e1ea33bb.jpg)
![](http://img.wesiedu.com/upload/2/aa/2aaa682e4eb673916289be49a13d4cd8.jpg)
![](http://img.wesiedu.com/upload/2/ea/2ea92571f601e66ebd0f5abfb358b685.jpg)
![](http://img.wesiedu.com/upload/f/53/f53518c209ea4ca8af46b66e0cab30b1.jpg)
![](http://img.wesiedu.com/upload/0/e3/0e3f14921f68eed22baab879202c350c.jpg)
设平面AMC的一个法向量为
![](http://img.wesiedu.com/upload/3/b7/3b7062c2d57c89efae52a1f3bcec4fce.jpg)
![](http://img.wesiedu.com/upload/d/3c/d3c6f0c2f1e8f97d655d3cc7d474a1bc.jpg)
令y="1," 则
![](http://img.wesiedu.com/upload/0/e3/0e3c3f56b0fa9dea54821febccb2a3f0.jpg)
![](http://img.wesiedu.com/upload/e/41/e41c657f52ec776830ee5b451d085b2f.jpg)
设平面ANC的法向量为
![](http://img.wesiedu.com/upload/8/79/8798b889b53e6333130fc99d6863aec1.jpg)
![](http://img.wesiedu.com/upload/6/a9/6a9def49354ee2216f18c3d55f01672b.jpg)
令y="1," 得平面ANC的一个法向量
![](http://img.wesiedu.com/upload/0/b6/0b635286adcacf32a43270f46e2448e0.jpg)
设二面角M-AC-N的平面角为
![](http://img.wesiedu.com/upload/2/df/2df12f7f3b01c6fa91f090da589881af.jpg)
![](http://img.wesiedu.com/upload/9/5e/95e5bc5ed7b002748da7cb9b7519281d.jpg)
(本题满分12分)如图,底面为菱形的四棱锥P-ABCD中,∠ABC=60°,AC="1," PA="2," PB=PD=
如图,在底面是菱形的四棱锥P-ABCD中,∠ABC=60°,PA=AC=a,PB=PD=根号2a,点E是PD的中点
如图,在底面是菱形的四棱锥P-ABCD中,∠ABC=60°,PA=AC=a,PB=PD=√2a,点E在PD上,且PE:E
如图,在底面是菱形的四棱锥P-ABCD中,∠ABC=60°,PA=AC=a,PB=PD= a,点E是PD的中点,
如图,在底面是菱形的四棱锥P-ABCD中,∠ABC=60°,PA=AC=a,PB=PD=a,点E在PD上,且PE:ED=
如图,在底面是菱形的四棱锥P-ABCD中,∠ABC=60o,PA=AC=a,PB=PD=√2a,点E在PD上,且PE:E
已知四棱锥P-ABCD的底面是边长为2的菱形,且∠ABC=60°,PA=PC=2,PB=PD.
在底面是菱形的四棱锥P-ABCD中,∠ABC=60°,PA=AC=a,PB=PD=√2a,点E在PD上,且BF:ED=2
1.在底面是菱形的四棱锥P-ABCD中,∠ABC=60°,PA=AC=a,PB=PD=根号2a,点E在PD上,且PE:E
如图:在底面为菱形的四棱锥P-ABCD中,PA=PC.PD=PB,点E是PD的中点.求证:AC垂直PB,PB平行面AEC
如图,在底面是菱形的四棱锥P—ABCD中,∠BDA=60°,PA=PD,E为PC的中点.(2)求证:PB⊥BC
已知四棱锥P-ABCD的底面ABCD为菱形,且∠ABC=60°,PB=PD=AB=2,PA=PC,AC与BD相交于点O.