1 . 《九章算术·商功》:“斜解立方,得两壍堵(qiàn dǔ).斜解壍堵,其一为阳马,一为鳖臑(biē nào).阳马居二,鳖臑居一,不易之率也.”文中所述可用下图表示:
![](https://img.xkw.com/dksih/QBM/2022/1/14/2894203289747456/2895176848777216/STEM/8c0d64b8-e798-4905-a031-747276d8fa23.png?resizew=384)
则几何体“鳖臑”的四个面中,直角三角形的个数为_______ ;若上图中的“立方”是棱长为1的正方体,则
的中点到直线
的距离等于________ .
![](https://img.xkw.com/dksih/QBM/2022/1/14/2894203289747456/2895176848777216/STEM/8c0d64b8-e798-4905-a031-747276d8fa23.png?resizew=384)
则几何体“鳖臑”的四个面中,直角三角形的个数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8772aa893a9c1d40f714cb25701701.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e539f26ed5e0b20ff7220559324869a4.png)
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解题方法
2 . 在四棱锥
中,底面
是正方形,
为棱
的中点,
,
且
,请求解下列问题.
(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/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db27b7f29d7d01b2692f217bc3079fc4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cbb05b8b630052ff544249ebd72d95d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f83a04565a8ebaa111894b724b0ba266.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/26/78e02653-a447-451b-abff-09504530d85a.png?resizew=152)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aebaf06bb1c96aecf49603c6a6bfcea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(3)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aebaf06bb1c96aecf49603c6a6bfcea.png)
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