【综合与实践】
在一次综合实践活动课上,张老师组织学生开展“如何仅通过折纸的方法来确定特殊平行四边形纸片一边上的三等分点”的探究活动.
【操作探究】
“求知”小组的同学经过一番思考和讨论交流后,对正方形
进行了如下操作:
第1步:如图1所示,先将正方形纸片
对折,使点A与点B重合,然后展开铺平,折痕为
;
第2步:将
边沿
翻折到
的位置;
第3步:延长
交
于点H,则点H为
边的三等分点.
证明过程如下:连接
,
∵正方形
沿
折叠,
∴
① ,
又∵
,
∴
,
∴
.
由题意可知E是
的中点,设
,则
,
在
中,可列方程:② ,(方程不要求化简)
解得:
③ ,即H是
边的三等分点.
“励志”小组对矩形纸片
进行了如下操作:
第1步:如图2所示,先将矩形纸片
对折,使点A与点B重合,然后展开铺平,折痕为
;
第2步:再将矩形纸片
沿对角线
翻折,再展开铺平,折痕为
,沿
翻折得折痕
交
于点G;
第3步:过点G折叠矩形纸片
,使折痕
.
(1)“求知”小组的证明过程中,三个空所填的内容分别是①: ,②: ,③: ;
(2)“励志”小组经过上述操作,认为点M为
边的三等分点,请你判断“励志”小组的结论是否正确,并说明理由.
【拓展提升】
(3)如图3,在菱形
中,
,E是
上的一个三等分点,记点D关于
的对称点为
,射线
与菱形
的边交于点F,请直接写出
的长.
在一次综合实践活动课上,张老师组织学生开展“如何仅通过折纸的方法来确定特殊平行四边形纸片一边上的三等分点”的探究活动.
【操作探究】
“求知”小组的同学经过一番思考和讨论交流后,对正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
第1步:如图1所示,先将正方形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
第2步:将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e5f736b1195fef1d2d300168a795f1.png)
第3步:延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
证明过程如下:连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6655e2fa64a32cd12fe0279afd65d73.png)
∵正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2a46af7768432787e743fc0545e4df6.png)
又∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1abd6283a5bee6f1b4dbb8e73468c956.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf266059d8fa8c1bd428bae7fceaad7.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f0b8af7ac91d87f4ca8d888dca74260.png)
由题意可知E是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69a0b211548844f12e6cab2575b24f66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7769e7570f8e20a395f9f974af3a4856.png)
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4575652a1b9a8b82b3b86549f20c7c05.png)
解得:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e09660e8bff0beac4eb769f8145e94b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
“励志”小组对矩形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
第1步:如图2所示,先将矩形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
第2步:再将矩形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.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/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
第3步:过点G折叠矩形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ff3c1ea812dbddf9c5752e45e3713a7.png)
(1)“求知”小组的证明过程中,三个空所填的内容分别是①: ,②: ,③: ;
(2)“励志”小组经过上述操作,认为点M为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
【拓展提升】
(3)如图3,在菱形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98b636cd30a364d6eda3ad54927c8a0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5b3bd5e6bc2a0a277d279bb01af9584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d226aef207ce71a381d6f63801cc9d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/557f8ff92d79a9d464ff13de17f3eae7.png)
更新时间:2024-06-24 15:09:57
|
相似题推荐
解答题-问答题
|
较难
(0.4)
【推荐1】探究1:
(1)如下图,在菱形
中,
点为射线
上一动点,
于
,连接
.当
时,
_________;
(2)如下图,在矩形
中,
为射线
上一点,
于
,连接
.当
时,
_________;
(3)如下图,在
中,
,
点为射线
上一点,
于
,连接
.(数据:
)
,则
____
;(填“>”或“=”或“<”)
②若
,求
的长.
(1)如下图,在菱形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c2b0d72881fca7e1e45114195ec13c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35eed9fb23f80002cc31dd57109e102a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eceff46380c92e4f8df5bce1a2bb061d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba4c15fb8fc3239d45bd4e7d8971f58e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4f76e37d6e6b182896ccaaa187cfe07.png)
(2)如下图,在矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2db444041879197f4a6d62f624c576ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35eed9fb23f80002cc31dd57109e102a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eceff46380c92e4f8df5bce1a2bb061d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba4c15fb8fc3239d45bd4e7d8971f58e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4f76e37d6e6b182896ccaaa187cfe07.png)
(3)如下图,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b38ffaad56fa8870336c833cef3788a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35eed9fb23f80002cc31dd57109e102a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eceff46380c92e4f8df5bce1a2bb061d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c10224d3c2d73fcfbb5c27ce36c93d08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d37abb3f11b8b09f669203b79da3cac7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d30b4015828f35d27adf7c12be8887a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e03cab2c9f78f55c14738123d22f3278.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba4c15fb8fc3239d45bd4e7d8971f58e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
您最近一年使用:0次
解答题-作图题
|
较难
(0.4)
【推荐2】在学完菱形后,某数学兴趣小组尝试利用手中的数学工具一三角板和圆规作出一个内角为60°的菱形,下面是他们探究过程中的讨论片段,请仔细阅读,并完成相应的任务.
小明:可以尝试利用含60°角的三角板和圆规作出菱形.如图,将三角极ABC放置在图纸上、延长直角边BA.以点C为圆心、CA长为半径作弧,以点A为圆心、AC长为半径作弧,交BA的延长线于点E,交上弧于点D,连楼CD,DE,则四边形ACDE即为所求作的菱形.
![](https://img.xkw.com/dksih/QBM/2022/3/25/2943702572048384/2965116421267456/STEM/c2e5cd78-77e6-4101-9706-8c7b4ddc689f.png?resizew=177)
小华:我可以在不利用三角板的前提下,作出符合要求的菱形,如图②,作半圆O及其直径AB、分到以点OB为圆心、大于
的长为半径作弧,两弧交于点MN,作直线MN交半圆O于点C;以点C为圆心、OC长为半径作弧,交半圆O于点D,连接AD,CD,CO,则四边形AOCD即为所求作的菱形.
![](https://img.xkw.com/dksih/QBM/2022/3/25/2943702572048384/2965116421267456/STEM/6e1483e2-6f4a-4b90-83ab-ed7f4211108c.png?resizew=199)
任务:
(1)小明的做法中,判断四边形ACDE是菱形的依据可能是______(填序号)
①四条边都相等的四边形是菱形 ②对角线互相垂直的四边形是菱形
③有一组邻边相等的平行四边形是菱形 ④对角线互相垂直的平行四边形是菱形
(2)你认为小华作出的四边形AOCD是有一个角为60°度的菱形吗?请判断并说理由.
(3)如图③,小齐利用含45°角的三角板ABC和圆规构造了菱形ABMN,已知点P是线段MC上的一个点,AB=10,当
时,请直接写出点P到直线MN的距离.
小明:可以尝试利用含60°角的三角板和圆规作出菱形.如图,将三角极ABC放置在图纸上、延长直角边BA.以点C为圆心、CA长为半径作弧,以点A为圆心、AC长为半径作弧,交BA的延长线于点E,交上弧于点D,连楼CD,DE,则四边形ACDE即为所求作的菱形.
![](https://img.xkw.com/dksih/QBM/2022/3/25/2943702572048384/2965116421267456/STEM/c2e5cd78-77e6-4101-9706-8c7b4ddc689f.png?resizew=177)
小华:我可以在不利用三角板的前提下,作出符合要求的菱形,如图②,作半圆O及其直径AB、分到以点OB为圆心、大于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/909a84c5f1e6ceea48418209d17c3a11.png)
![](https://img.xkw.com/dksih/QBM/2022/3/25/2943702572048384/2965116421267456/STEM/6e1483e2-6f4a-4b90-83ab-ed7f4211108c.png?resizew=199)
任务:
(1)小明的做法中,判断四边形ACDE是菱形的依据可能是______(填序号)
①四条边都相等的四边形是菱形 ②对角线互相垂直的四边形是菱形
③有一组邻边相等的平行四边形是菱形 ④对角线互相垂直的平行四边形是菱形
(2)你认为小华作出的四边形AOCD是有一个角为60°度的菱形吗?请判断并说理由.
(3)如图③,小齐利用含45°角的三角板ABC和圆规构造了菱形ABMN,已知点P是线段MC上的一个点,AB=10,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/319b1b275befb2dfc896574ab9d8b51b.png)
![](https://img.xkw.com/dksih/QBM/2022/3/25/2943702572048384/2965116421267456/STEM/ce06caaa-ea95-4251-aa52-d27ad491ca12.png?resizew=245)
您最近一年使用:0次
解答题-证明题
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较难
(0.4)
【推荐1】如图1,在正方形
中,点E是
边上的一点,
,且
交正方形外角的平分线
于点P.
(1)求
的度数;
(2)求证:
;
(3)在
边上是否存在点M,使得四边形
是平行四边形?若存在,请画出图形并给予证明;若不存在,请说明理由;
(4)如图2,在边长为4的正方形
中,将线段
沿射线
平移,得到线段
,连接
,则直接写出
的最小值是 .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b78356dbceb326e1b1561e0a15d8ebb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f541f7ae7c39082d202efd28805c54e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63a253c7fdf589ee3dece13d5b5b5732.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/10/70587db9-f7a2-4c5f-8363-135603e537b5.png?resizew=342)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d451070229ac0bbba9fe73dbd9600602.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcd72a32b01cde3803f106f00a83458c.png)
(3)在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bab17eb0eb1d6c18adeccdef2b62c826.png)
(4)如图2,在边长为4的正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0e57a13c665af88f326c9890072bf73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4126c5be43de8a0299952b40438a287.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/115c19e4b1f2524ec25a6d7fac1b837b.png)
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解答题-证明题
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(0.4)
名校
【推荐2】正方形ABCD的边长为6,点E是BC边上一动点,点F是CD边上一动点,过点E作AF的平行线,过点F作AE的平行线,两条线交于点G.
![](https://img.xkw.com/dksih/QBM/2022/2/10/2913635960414208/2987560027086848/STEM/aba83818-ac0b-442e-80a6-efc450cafe1b.png?resizew=581)
(1)如图1,若BE=DF,求证:四边形AEGF是菱形;
(2)如图2,在(1)小题条件下,若∠EAF=45°,求线段DF的长;
(3)如图3,若点F运动到DF=2的位置,且∠EAF依然保持为45°,求四边形AEGF的面积.
![](https://img.xkw.com/dksih/QBM/2022/2/10/2913635960414208/2987560027086848/STEM/aba83818-ac0b-442e-80a6-efc450cafe1b.png?resizew=581)
(1)如图1,若BE=DF,求证:四边形AEGF是菱形;
(2)如图2,在(1)小题条件下,若∠EAF=45°,求线段DF的长;
(3)如图3,若点F运动到DF=2的位置,且∠EAF依然保持为45°,求四边形AEGF的面积.
您最近一年使用:0次
解答题-证明题
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【推荐1】在矩形ABCD中,BC=12,对角线AC、BD相交于点O,∠AOB=α,点P为对角线BD上一动点,且与B、O、D三点不重合,连接AP.
![](https://img.xkw.com/dksih/QBM/2019/11/13/2333040747913216/2333190706798592/STEM/23c65772569e44219a2c1c8eed29b545.png?resizew=485)
(1)如图1,当点P在BO上时,将AP绕点P逆时针方向旋转α得到EP,连接BE;过点P作PF∥OA,交AB于点F
①判断AF与BE的数量关系,并证明你的结论;
②当α=55°时,求∠ABE的度数.
(2)将AP绕点P顺时针方向旋转(180°﹣α)得到EP,连接DE,当DP=3OP时,请你在备用图中画出图形,并求出DE的长度.
![](https://img.xkw.com/dksih/QBM/2019/11/13/2333040747913216/2333190706798592/STEM/23c65772569e44219a2c1c8eed29b545.png?resizew=485)
(1)如图1,当点P在BO上时,将AP绕点P逆时针方向旋转α得到EP,连接BE;过点P作PF∥OA,交AB于点F
①判断AF与BE的数量关系,并证明你的结论;
②当α=55°时,求∠ABE的度数.
(2)将AP绕点P顺时针方向旋转(180°﹣α)得到EP,连接DE,当DP=3OP时,请你在备用图中画出图形,并求出DE的长度.
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名校
【推荐2】已知抛物线y=ax2+ c(a≠0).
(1)若抛物线与x轴交于点B(4,0),且过点P(1,–3),求该抛物线的解析式;
(2)若a>0,c =0,OA、OB是过抛物线顶点的两条互相垂直的直线,与抛物线分别交于A、B 两点,求证:直线AB恒经过定点(0,
);
(3)若a>0,c <0,抛物线与x轴交于A,B两点(A在B左边),顶点为C,点P在抛物线上且位于第四象限.直线PA、PB与y轴分别交于M、N两点.当点P运动时,
是否为定值?若是,试求出该定值;若不是,请说明理由.
(1)若抛物线与x轴交于点B(4,0),且过点P(1,–3),求该抛物线的解析式;
(2)若a>0,c =0,OA、OB是过抛物线顶点的两条互相垂直的直线,与抛物线分别交于A、B 两点,求证:直线AB恒经过定点(0,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/476163ae8bff995a4d33c367b6beab67.png)
(3)若a>0,c <0,抛物线与x轴交于A,B两点(A在B左边),顶点为C,点P在抛物线上且位于第四象限.直线PA、PB与y轴分别交于M、N两点.当点P运动时,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14c75ad7898c8c6fff50a167624af746.png)
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【推荐1】如图1,在平面直角坐标系中,有一张矩形纸片
,已知
,
,
,点P是
边上的动点(与点O、A不重合).现将
沿
翻折,得到
;再在
边上选取适当的点E,将
沿
翻折,得到
,并使直线
、
重合.
(1)设
,
,求y关于x的函数关系式,并求y的最大值;
(2)如图2,若翻折后点D落在
边上,求过点P、B、E的抛物线的函数关系式;
(3)在(2)的情况下,在该抛物线上是否存在点Q,使
是以
为直角边的直角三角形?若不存在,说明理由;若存在,求出点Q的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3241d7fedd89d85711acd7a2635298af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c432be73ca47930da14921ecd860ba67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6109d39796845c833c05c4708951d53a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/922d53ce802eeed79fd08e1b1f10e298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0bb4ef08f6070a6bbc54ea9c4f130d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08b98bec817e96fba365ba496ad3b9af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc5adb5eb60ae4435a12d93854066298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/def9265137e5fd300081684e44f4ceb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fb26d84907c923278ac4626a9d58947.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/18/ca1b1d5c-9e43-40ab-969c-5a9e44a0ca86.png?resizew=330)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89c03ccc05d4592aae25b8e8fa1505e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8904130325ed9c28dc7d529435c038b.png)
(2)如图2,若翻折后点D落在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(3)在(2)的情况下,在该抛物线上是否存在点Q,使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e310ac26049fd114f77837f60938a17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc5adb5eb60ae4435a12d93854066298.png)
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【推荐2】如图.抛物线
与y轴交于点
,与x轴交于A,B两点,A点在对称轴的左侧,B点的坐标为
.
(2)设抛物线的对称轴与直线
交于点D,连接
,
,求
的面积;
(3)点E为直线
上一动点,过点E作y轴的平行线与抛物线交于点F,是否存在点E,使得以点D,E,F为顶点的三角形与
相似?若存在,求出点E的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22e59da5115d0dafea24822245f92c48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/850e01debd266a398b597a1876dbb3ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb442fb1fe1a5d9768cf11c3e1b7ec5b.png)
(2)设抛物线的对称轴与直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
(3)点E为直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff36ac07c45c8d03cf3b7d8739c517ca.png)
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【推荐3】【问题初探】
(1)在数学活动课上,李老师给出如下问题:
如图1,在
中,
,
.点
,
在
上,且
,探究线段
的数量关系.
如图2,小鹏在探索过程中发现,这三条线段的数量关系不是简单的和或差的关系,于是就考虑能否将这三条线段转化到同一个三角形中,进而探索其数量关系,联想到近期学的旋转变换,小鹏同学给出如下探索思路:过点C作
的垂线,并在垂线上截取
,连接
,通过证明三角形全等,得出对应的线段相等,进而将线段
的数量关系转化为
与
的数量关系.请你根据小鹏的解题思路,写出证明过程.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/4/0e9e04c4-120a-4a44-85d5-bb71c3b509f9.png?resizew=354)
【学以致用】
(2)如图3,在四边形
中,
,
,
,
,求
的长.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/3/8318127b-3a49-4fab-bd45-54e2c6ab52a2.png?resizew=160)
(1)在数学活动课上,李老师给出如下问题:
如图1,在
![](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/4b10134e7a46e6f6f7cb9d5e2371727d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a1af32e267e35ef79bcb8df5d169283.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/accab5af6333cc2690cf08cd40f6b200.png)
如图2,小鹏在探索过程中发现,这三条线段的数量关系不是简单的和或差的关系,于是就考虑能否将这三条线段转化到同一个三角形中,进而探索其数量关系,联想到近期学的旋转变换,小鹏同学给出如下探索思路:过点C作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aebb5baea4ac1668b674a8634e15bcbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5912e3dd19cd8024dfb950bad0358a11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/accab5af6333cc2690cf08cd40f6b200.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de7cb98ebad4ce465a7c02fc1c3b5e0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/4/0e9e04c4-120a-4a44-85d5-bb71c3b509f9.png?resizew=354)
【学以致用】
(2)如图3,在四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/181b71ef9fc4691a765c9605fd76288b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/669f7a1e43ef893ef3a45fc47992947f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07160f14b3b453bebb64cb2bf96dc85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e673ef2d48215ca84a48377f17d6df00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/3/8318127b-3a49-4fab-bd45-54e2c6ab52a2.png?resizew=160)
图3
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