解题方法
1 . 已知,在半圆
中,直径
,点C,D在半圆O上运动,弦
.
(1)如图1,当
时,求证:
;
(2)如图2,若
,求图中阴影部分(弦
、直径
、弧
围成的图形)的面积;
(3)如图3,取
的中点M,点C从点A开始运动到点D与点B重合时结束,在整个运动过程中:点M到
的距离的最小值是______.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc34db5860990e51ba31edc8cdd077c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b515965c22d2950b592c096c6e3bdfd4.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/13/3799d421-8493-41f1-bcae-b27641617cac.png?resizew=465)
(1)如图1,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bb12d9cbf4d45e8efc4f53c41e2d932.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/612c9971e6128fe47301fb9fd26ba37d.png)
(2)如图2,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8b1a6f76e8d9425c4c904022717c1bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
(3)如图3,取
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
您最近一年使用:0次
2 . 如图,以
为直径的半圆O交
于点B,以点B为圆心,
长为半径的半圆B过点A与点O,若
,则阴影部分的面积是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf9ad150cb1e4cd8977d4cc3d99be17c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/2/be135902-1b58-4410-b8e1-e0a9badb7cdb.png?resizew=213)
您最近一年使用:0次
3 . 在数学活动课中,数学学习小组对课本习题进行探究.
已知:
和
外一点
.你能用尺规过点
作
的切线.
小明的作法:
①连接
,如图
;
②分别以
和点
为圆心,大于
的长为半径作弧,两弧相交于
,
两点;
③作
,交
于点
;
④以点
为圆心,
的长为半径作圆,交
于
,
两点;
⑤作直线
,
.直线
,
即为
的切线.
(1)请结合上述作图步骤,证明小明所作的直线为
的切线.
(2)已知
,如图3,连接
,
,
交
于点
,连接
,
.
①当
_______时,四边形
是正方形;
②若
,求阴影部分的面积.
已知:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/10/d667e445-8af2-467f-a2d5-39a56977e230.png?resizew=372)
小明的作法:
①连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
②分别以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b64dc1ab9bf65cfdaac7633e9a7958ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
③作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
④以点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5a122e25cf4eb9f03ffe5ec823bfc31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
⑤作直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
(1)请结合上述作图步骤,证明小明所作的直线为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d3c34fe8c99745f61e57db23bbdb1e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/10/00b05d1f-357d-4e6b-aff3-5e3e19f50ddb.png?resizew=188)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f75158101661ac908d605abfe2048226.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/936cf05b8228b6eeeb7bfeae811d6a34.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de853a263e59781532f89fdab2a0acb7.png)
您最近一年使用:0次
名校
4 . 如图,在扇形
中,
,点C为
的中点,点D为
的中点,连接
,交
于点E,若
,则阴影部分的面积是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274a77343ecde1c2665df291761b6563.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ccc37b189fa2cbc269ca0b233dac37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d65cecaf8a3dc2953f4109c75a981e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9774f83067ed956a551bc41adcce0469.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/28/09763f9a-a971-4673-a67b-5088e9fa110a.png?resizew=146)
您最近一年使用:0次
2023-05-24更新
|
250次组卷
|
2卷引用:2023年河南省许昌市中考二模数学试题
5 . 在平面内,将小棒
经过适当的运动,使它调转方向(调转前后的小棒不一定在同一条直线上),那么小棒扫过区域的面积如何尽可能地小呢?
已知小棒长度为4,宽度不计.
方案1:将小棒绕
中点O旋转180°到
,设小棒扫过区域的面积为
(即图中灰色区域的面积,下同);
方案2:将小棒先绕A逆时针旋转60°到
,再绕C逆时针旋转60°到
,最后绕B逆时针旋转60°到
,设小棒扫过区域的面积为
.
(1)①
______,
______;(结果保留
)
②比较
与
的大小.(参考数据:
,
.)
(2)方案2可优化为方案3:首次旋转后,将小棒先沿着小棒所在的直线平移再分别进行第2、3次旋转,三次旋转扫过的面积会重叠更多,最终小棒扫过的区域是一个等边三角形.
①补全方案3的示意图;
②设方案3中小棒扫过区域的面积为
,求
.
(3)设计方案4,使小棒扫过区域的面积
小于
,画出示意图并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
已知小棒长度为4,宽度不计.
方案1:将小棒绕
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c69596e1040b4b35b7c1e05e2533cc99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
方案2:将小棒先绕A逆时针旋转60°到
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c69596e1040b4b35b7c1e05e2533cc99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/29/a12cae96-da4b-4e58-8065-1b8155c20895.png?resizew=529)
(1)①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3dabace330ccdd9755fbcf32d5d6343.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1b625192d7398634a02ea24fa78ed87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86ebba6ed1add0fe647c0226614b9290.png)
②比较
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a061dce0c7716ceab53159cd4ce7904.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/460317e7c26f95b9b29cfe1a89b796d6.png)
(2)方案2可优化为方案3:首次旋转后,将小棒先沿着小棒所在的直线平移再分别进行第2、3次旋转,三次旋转扫过的面积会重叠更多,最终小棒扫过的区域是一个等边三角形.
①补全方案3的示意图;
②设方案3中小棒扫过区域的面积为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
(3)设计方案4,使小棒扫过区域的面积
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f30f56664446f32dbbc2c5f12a99374.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
您最近一年使用:0次
名校
6 . 如图,
、
是
中关于直径
对称的两条弦,以弦
、
为折线将弧
,弧
折叠后过圆心O,若
的半径
,则圆中阴影部分的面积为______ .
![](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/3d97cdc586744d208b6f69c9813af977.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/03902478df1a55bc99703210bccab910.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/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a5900719c2da83d76bef8c7a0abdf82.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/26/94cd908c-703c-48e5-96fd-be8e1784455f.png?resizew=145)
您最近一年使用:0次
名校
7 . 如图所示,
,
,将扇形
绕边
的中点D顺时针旋转
得到扇形
,弧
交
于点E,则图中阴影部分的面积为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ccc37b189fa2cbc269ca0b233dac37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32268e2ccf8f7bc6fe5787ec49958a6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4819c39c281427826e1b3f7a4c2b720.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22a633df6d1b15deeab2acd9040c7177.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb0628cecbfc98d390e5447d52414e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/20/c9e5ca68-8857-4fde-a116-a46ee8c5ba9e.png?resizew=158)
您最近一年使用:0次
8 . 如图,是
的外接圆,
,点D是
的中点,连接
,过点A作
的垂线交
的延长线于点E,连接
并延长
与
的延长线交于点F.
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fc56c77464a17a1e97b568762a3e2c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31320a646b1734d727e466ceadfc2d59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
(3)在(2)的条件下,求阴影部分的面积.
您最近一年使用:0次
2023-05-16更新
|
279次组卷
|
2卷引用:2023年贵州省仁怀市中考二模数学试题
9 . 如图所示,
为矩形,以
为直径作半圆,矩形的另外三边分别与半圆相切,沿着折痕
折叠该矩形,使得点
的对应点
落在
边上,若
,则图中阴影部分的面积为__________________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/09d27bd71d79cb19eb554175e4ef0867.png)
![](https://img.xkw.com/dksih/QBM/2023/5/10/3234706473902080/3238307681075200/STEM/5a4a34f419af418c950a7c58dc8bdfaa.png?resizew=276)
您最近一年使用:0次
2023-05-15更新
|
193次组卷
|
8卷引用: 2023年山东省青岛市中考数学一模试题
2023年山东省青岛市中考数学一模试题2023年山东省青岛市市南区教研联合体中考一模数学试题2023年河南省洛阳市洛宁县中考一模数学试题2023年河南省洛阳市洛宁县中考第三次大练习(一模)数学试题四川省自贡市富顺县六校联考2019届中考三模数学试卷2019年河南省重点中学内部模拟数学(二)试题2021年广东省深圳市中考数学冲刺模拟试卷(一)(已下线)2022年广东省佛山市南海市狮山镇桃园初级中学九年级数学中考复习第一次模拟测试题
10 . 如图,在
中,
,
平分
交
于点E,点O在
上,经过点A,E的
分别交
于点D,F,连接
交
于点P.
(1)求证:
是
的切线;
(2)若
,求阴影部分的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60d9142db4dd2ef151bf3d4a63afb61e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](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/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5dc62e10004e73908091338362917da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/28/685e7c62-f0c6-4420-ab8c-987141711e03.png?resizew=174)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f46dc69ca35e2f82b665ed0c36f275c1.png)
您最近一年使用:0次