某班“数学兴趣小组”在学习“勾股定理”章节的内容后,遇到这样的问题:如图,在直角三角形ACB中,
,
,点D是边CB上的一个动点(不与B、C重合),连接AD.若
是等腰三角形,求线段CD的长.
![](https://img.xkw.com/dksih/QBM/2021/6/11/2740712708087808/2741356457312256/STEM/d4fef249-1758-4f79-a28b-09575c420205.png)
方法一:小敏利用刚学习的勾股定理进行解决,当
为等腰三角形时,
,设
,则
,所以
,在直角三角形ACD中,利用勾股定理可得,
,
解得
.故当
为等腰三角形时,CD的长为
.
方法二:小聪提前预习了函数这一章节的内容,他尝试利用函数的方法探究并解决该问题.
下面是他的探究讨程,请你补充完整.
(1)根据点D在PC上的不同付置,画出相应图形,测量出线段CD、AD的长度,得出下面的表格:
①表格中
的值为_____________.
②小聪分析得知不用测量BD的值,因为CD与BD满足关系式:___________.
(2)将CD的长作为自变量x,AD的长为x的函数,记为y,在下面平面直角坐标系中画出函数y关于x的图象,并写出该函数的一条性质:_________________________________.
(3)继续在平面直角坐标系画出小聪所需的其他函数图象,并结合图形直接写出,当
为等腰三角形时,线段CD的长度的近似值(精确到0.1).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e4cec8a7b87c2bc9fb135eadb7aa5c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a81c5699a01b6b12b16ff9bce6d6ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5f517953a21c2a45fd8465072c44bfc.png)
![](https://img.xkw.com/dksih/QBM/2021/6/11/2740712708087808/2741356457312256/STEM/d4fef249-1758-4f79-a28b-09575c420205.png)
方法一:小敏利用刚学习的勾股定理进行解决,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ff29971ccc633d89832ffa9bd54afa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd4a7d7c2253a1686c61f8b8f3f6c6a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/836c5914d8c1128ffee91d3cd5a14d37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e3527842f665d85e4754b9147506d4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24c81a92f19f9c1b59c8df5e5be6aa47.png)
解得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8279140f3a6eb1e7171e524403c07876.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d297eab7380f6a28ec010218d9ab4ba1.png)
方法二:小聪提前预习了函数这一章节的内容,他尝试利用函数的方法探究并解决该问题.
下面是他的探究讨程,请你补充完整.
(1)根据点D在PC上的不同付置,画出相应图形,测量出线段CD、AD的长度,得出下面的表格:
CD | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
AD | 6 | 6.1 | 6.3 | 6.7 | 7.2 | 7.8 | 8.5 | 9.2 | a |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
②小聪分析得知不用测量BD的值,因为CD与BD满足关系式:___________.
(2)将CD的长作为自变量x,AD的长为x的函数,记为y,在下面平面直角坐标系中画出函数y关于x的图象,并写出该函数的一条性质:_________________________________.
(3)继续在平面直角坐标系画出小聪所需的其他函数图象,并结合图形直接写出,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://img.xkw.com/dksih/QBM/2021/6/11/2740712708087808/2741356457312256/STEM/3e2f4292-0092-4f6a-b8bf-f56472941e6d.png)
更新时间:2021-06-12 12:46:54
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【推荐1】已知A、B两地相距240km,甲开汽车从A地到B地出差,甲出发1小时后,乙开货车装满货物从B地驶往A地,图中两条线段分别表示甲乙两车与B地的距离S(km)与行驶时间t(h)的变量关系;请根据以上信息结合图象回答以下问题:
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【推荐2】甲、乙两人参加从M地到N地的长跑比赛,两人在比赛时所跑的路程
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【推荐1】在初中阶段的函数学习中,我们经历了列表、描点、连线画出函数图象,并结合图象研究函数性质及其应用的过程.同时,我们也学习了绝对值的意义:
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(1)根据表格填空:
__________,
__________,自变量
的取值范围是__________;
(2)在平面直角坐标系中,画出函数图象,并根据这个函数的图象,写出该函数的一条性质;
(3)已知函数
的图象如图所示,结合你所画的函数图象,直接写出不等式
的解集.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ce293fdbc5745a87c0296c682a21fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/219226486aa05501223604b721240a62.png)
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![]() | … | -6 | -5 | -4 | -3 | -2 | 0 | 1 | 2 | 3 | 4 | … |
![]() | … | ![]() | 3 | ![]() | 4 | ![]() | 2 | 0 | ![]() | 1 | ![]() | … |
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【推荐2】为解方程
,小舟根据学习函数的经验对其进行了探究,下面是其探究的过程,请补充完整:
(1)先研究函数
,列表如表:
表格中,m的值为__________.
(2)如图,在平面直角坐标系
中,描出了以上表中各组对应值为坐标的点,根据描出的点,画出了函数
图象的一部分,请根据剩余的点补全此函数图象.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/7/d40233a5-c55a-4284-b9e1-1c59eb4f104f.png?resizew=199)
(3)观察图象,当
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(4)在第(2)问的平面直角坐标系中画出直线
.根据图象直接写出方程
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecf82d3a0246c855b64184adceffc3a7.png)
x | ![]() | ![]() | 0 | ![]() | 1 | 2 | ![]() |
y | 0 | ![]() | 0 | m | ![]() | 0 | ![]() |
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【推荐1】在
中,
,D为
边上一点,但不与点A、点C重合,过点D作
于点E,连接
,F为
的中点,连接
、
.
![](https://img.xkw.com/dksih/QBM/2022/1/20/2898669644013568/2899175009173504/STEM/d3baf476-7715-4aea-a4e8-b1b3f91ea367.png?resizew=424)
(1)如图1,求证:
;
(2)如图2,将
绕点A顺时针旋转,其他条件不变,则(1)中的结论还成立吗?若成立,请写出证明过程,若不成立,请说明理由;
(3)若
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,
,在
绕点A旋转一周的过程中,当直线
经过点B时,求线段
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(1)如图1,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efbc0e76b7abb656327bdc97226713e1.png)
(2)如图2,将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab926d89b65f26c12e3da73ef1e5cf68.png)
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【推荐1】某数学活动小组在一次活动中,对一个数学问题做如下研究:
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/2/1ce48cbb-7e4c-4f12-9c93-2c2b7c338acb.png?resizew=561)
(1)如图1,在
中,分别以
,
为边向外作等腰
和等腰
,使
,
,
,连接
,
,试猜想
与
的大小关系,并说明理由;
(2)如图2,在
中,分别以
,
为边向外作等腰直角三角形
和等腰直角三角形
,
,连接
,
,若
,
,
,求
的长.
(3)如图3,在四边形
中,连接
,
,
,
,
,
,求
的长 (画出图形,做必要标记,不必写过程).
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/2/1ce48cbb-7e4c-4f12-9c93-2c2b7c338acb.png?resizew=561)
(1)如图1,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca09b202735587ec2df48681c62f7382.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fc13fe21e64d9b45614ed43be847904.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02fbbe04651b37e1520bb9a7bafed509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
(2)如图2,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09afc70f448545336304333d5b5658b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7013e123fb2dde7ee86f0374846cb1b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b3a593f7126c0f4629df5ae20c5500a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3098d1599f9b5bd18208db5fdac4f117.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c42ed2e5bd5a0f033e24008697bf4963.png)
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(3)如图3,在四边形
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19a1bf64ba772922fe58b5322eb4bced.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00ec435aa1401dbce7863b531bf2f3e5.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dd90bfa2987b84df430498021d5f648.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2f5ae7930c508d327a7609f4cf79b61.png)
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名校
【推荐2】模型建立:
如图1,在等腰直角
中,
,
,直线
经过点
,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/11/2d82a563-35af-4451-9847-70c2675917ed.png?resizew=497)
模型应用:
(1)求证:
;
(2)已知直线
:
与
、
轴分别交于点
、
,直线
过点
,且与
的夹角等于
,如图
,求直线
的函数表达式.;
(3)如图3,在长方形
中,点
,点
是线段
上一动点,
,已知点
在第 一象限,是直线
上的一点,若
是等腰三角形,且
,请直接写出点
的坐标.
如图1,在等腰直角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82da6815bc213dfd78c2f77cd7ded8ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92b9fe03c2f180e88637e47b4f6eda88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8946331b0a9d86e1a9c78797f3021455.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ae3334667a907e211a369d1457f861e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6749f967f0e002d91207e4f48a46ca13.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/11/2d82a563-35af-4451-9847-70c2675917ed.png?resizew=497)
模型应用:
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/347035600b2754641d2e9e7c1629fa52.png)
(2)已知直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5d27c081879b6c6bc0abb0ceedb9d3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8a98e24892abc0eb34d5216d4e89c61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60d45e12a06ff0d0a6a53f2bfe573bc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b342d5cb6dd1c84a82f6aca68ed829c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60d45e12a06ff0d0a6a53f2bfe573bc2.png)
(3)如图3,在长方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84f17a3fa4036efe1d4c35c14e57952e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b271908fb7902911fe09949604e569ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764509115979e9958101808383672ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/073d9218f4a6838a67143ce6a3504d7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa3289d35168114842edaa9ab429b189.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ad76cea565f95292dcdfd6b8dc0e73a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31dba6c2918616aca7590f2c8e33ac04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
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