1 . 某数学兴趣小组对一个数学问题的探究过程如下,请仔细阅读,并解答相应问题.
【问题】如图,
中,
,
cm,D为
边上一个动点,连接
,过点C作
,垂足为点E,F为线段
上一点,且
,过点F作
交直线
于点G,判断线段
,
,
的数量关系.
【观察】数学兴趣小组的同学观察到线段
,
,
的长度随
的长度变化而变化,但他们并没有发现明显规律.
【实验】他们借助电脑软件根据点D在
上的不同位置,测量线段
,
,
,
的长度,得到下表的几组对应值.
请根据以上信息,完成下列问题.
【猜想】(1)线段
,
,
的数量关系为____.
【证明】(2)请证明上述猜想.
【拓展】(3)上述问题中,若D为射线
上的一个动点,F为射线
上的一个动点,其他条件不变,当
cm时,直接写出
的长.
【问题】如图,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed10df4140819d5451773a45de66201b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f13e584349ba3775c49424fe646d8460.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/497846628a41a9bc750a645e045afb47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1642eec556eb252de9c1ab7bb5ca90b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88bbc115577cd400ad9b83b0102440c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a946f9010d77f54d566e338ee13b76a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9abaeba15f3abdd877bc701af52c5cd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0e57a13c665af88f326c9890072bf73.png)
【观察】数学兴趣小组的同学观察到线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0e57a13c665af88f326c9890072bf73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
【实验】他们借助电脑软件根据点D在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0e57a13c665af88f326c9890072bf73.png)
![]() | 1.00 | 2.00 | 3.00 | 4.00 | 5.00 |
![]() | 0.98 | 1.85 | 2.57 | 3.12 | 3.54 |
![]() | 0.20 | 0.74 | 1.54 | 2.50 | 3.54 |
![]() | 0.78 | 1.11 | 1.03 | 0.62 | 0 |
【猜想】(1)线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0e57a13c665af88f326c9890072bf73.png)
【证明】(2)请证明上述猜想.
【拓展】(3)上述问题中,若D为射线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1642eec556eb252de9c1ab7bb5ca90b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37838f9e82d7fcebfd8ec0f192dfab9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/3/2a701aea-d86a-45eb-bd82-7bfcb781ea20.png?resizew=280)
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