如图,在平行四边形ABCD中,BC=AC,E、F分别是AB、CD的中点,连接CE、AF.
(1)求证:四边形AECF是矩形;
(2)当平行四边形ABCD的边或角满足什么关系时,四边形AECF是正方形?请说明理由.
(3)在(2)的条件下,若AE=4,点M为EC中点,当点P在线段AC上运动时,求PE+PM的最小值.
(1)求证:四边形AECF是矩形;
(2)当平行四边形ABCD的边或角满足什么关系时,四边形AECF是正方形?请说明理由.
(3)在(2)的条件下,若AE=4,点M为EC中点,当点P在线段AC上运动时,求PE+PM的最小值.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/13/651c638e-a801-4a5c-82ce-3efc3ccbd7f2.png?resizew=219)
更新时间:2021-08-14 18:00:15
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【知识点】 四边形中的线段最值问题解读
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解答题-证明题
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【推荐1】如图,在Rt△ABC中,∠ACB=90°,AC=4,BC=6,点E是斜边AB上的一个动点,连接CE,过点B,C分别作BD∥CE,CD∥BE,BD与CD相交于点D.
(1)当CE⊥AB时,求证:四边形BECD是矩形;
(2)填空:
①当BE的长为______时,四边形BECD是菱形;
②在①的结论下,若点P是BC上一动点,连接AP,EP,则AP+EP的最小值为______.
(1)当CE⊥AB时,求证:四边形BECD是矩形;
(2)填空:
①当BE的长为______时,四边形BECD是菱形;
②在①的结论下,若点P是BC上一动点,连接AP,EP,则AP+EP的最小值为______.
![](https://img.xkw.com/dksih/QBM/2020/3/3/2411446934274048/2411688985165825/STEM/8b3d48723d55417c99631c2c459cca61.png?resizew=140)
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解答题-问答题
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真题
【推荐2】如图一,在射线
的一侧以
为一条边作矩形
,
,
,点
是线段
上一动点(不与点
重合),连结
,过点
作
的垂线交射线
于点
,连接
.
![](https://img.xkw.com/dksih/QBM/2019/7/26/2255226562363392/2255863090970624/STEM/545be3b65be84e87aaf4ffa31dfad35d.png?resizew=420)
(1)求
的大小;
(2)问题探究:动点
在运动的过程中,
①是否能使
为等腰三角形,如果能,求出线段
的长度;如果不能,请说明理由.
②
的大小是否改变?若不改变,请求出
的大小;若改变,请说明理由.
(3)问题解决:
如图二,当动点
运动到
的中点时,
与
的交点为
,
的中点为
,求线段
的长度.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7785afeeaf274892253d04b4f693b367.png)
![](https://img.xkw.com/dksih/QBM/2019/7/26/2255226562363392/2255863090970624/STEM/545be3b65be84e87aaf4ffa31dfad35d.png?resizew=420)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5157b42da58d55daad27d98b2fec15ff.png)
(2)问题探究:动点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
①是否能使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5a9fc2dda89829fa2c610a8fce86d22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c884b508394b3ab50734b584d9ec783c.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e44d9f2b82b73e7a0917a19f74dbc5f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e44d9f2b82b73e7a0917a19f74dbc5f9.png)
(3)问题解决:
如图二,当动点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83f1f880e5ffbff036953acaca90c41.png)
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