1 . 已知
,其中
,则
的值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ada395205403b7dacb6b6317d094dee9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b22aeac7621390972a2ad508534290ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bab46e06bfa4bd3adabcd0f73fb1e268.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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2 . 化简:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84c77eb5cec923c688dc62b1e40ed305.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84c77eb5cec923c688dc62b1e40ed305.png)
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3 . 如图,抛物线
与
轴交于
,
两点,与
轴交于点
.
(1)求抛物线的解析式;
(2)点
是抛物线上一动点.
①当
时,求点
坐标;
②如图2,当点
运动到抛物线的顶点时,作
于点
,点
在直线
上,点
在平面内,若以
,
,
,
为顶点的四边形是矩形,请直接写出点
的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58d88bbd34102b55fa928e8ff83f0d52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0929421a6188c3122442866b0b85a5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08ef03f452410ab19c6246567c427178.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/19/7b11721b-5e64-402c-b98a-666019ac1bca.png?resizew=468)
(1)求抛物线的解析式;
(2)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e93e1661d70eca6d7eb848b667e388e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
②如图2,当点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/545e18836bc7fee22f8f813a6f525d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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4 . 如图,
的直径
与弦
交于点E,已知
,
,
,那么
的值为_______ .
![](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/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0e0638c2cee8e2fd2d1149061a8c275.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eab2ccc075e9eb14adcaad5feaf69896.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30455a77001ddc5c1336f87647849ec5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5a18fd99d731314595558f89f5f4aa8.png)
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名校
5 . 如图,在平面直角坐标系中,菱形ABCD的顶点D在第二象限,其余顶点都在第一象限,
轴,
,
.过点A作
,垂足为E,
.反比例函数
的图象经过点E.与边AB交于点F,连接OE,OF,EF.则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/763c3cc84f49dfc58350fe4a38ee0741.png)
______ .若
,则点F坐标为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc195a9f7ac138e5a1ad87d5fa201c49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0ee3e8e5fc6c302b8cdf69792d88e31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47dfbfbfadf86c6ca8363a93b5157c99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4611ceb2a28f7a7e4d24266d7f99b22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f684d668177d0363cabbcd8e1e7b5c2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9212e728b36c078188606c9d429389d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/763c3cc84f49dfc58350fe4a38ee0741.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/747f9d2a9389f6633d128c1764bde88e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/6/30/21ab0ee1-baaf-4cd4-ac57-73a4222ae9d3.png?resizew=165)
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6 . 如图,抛物线
与
轴交于A,
两点,且
,与
轴交于点
,连接
,抛物线对称轴为直线
,
为第一象限内抛物线上一动点,过点
作
于点
,与
交于点
,设点
的横坐标为
.
![](https://img.xkw.com/dksih/QBM/2022/4/24/2964951763935232/2967380065320960/STEM/f7f5e97c8f6442d29d809c37180c7c68.png?resizew=347)
(1)求抛物线的表达式;
(2)当线段
的长度最大时,求
的值:
(3)点
是抛物线对称轴上的一点,点
是坐标平面内的一点,是否存在点
,使得以点
,
,
,
为顶点的四边形是菱形,若存在,请直接写出点
的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a0d44326c214c7a5ecc3068a29ec96f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3abef063e1692494fd4769d100f044b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b650820d7bed48ed67a2869ad8c65ff1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/504cf35247693ceb766c3b85a4ab2a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://img.xkw.com/dksih/QBM/2022/4/24/2964951763935232/2967380065320960/STEM/f7f5e97c8f6442d29d809c37180c7c68.png?resizew=347)
(1)求抛物线的表达式;
(2)当线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2db25f96950925b1cc1a82e05fd28f5e.png)
(3)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.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/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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名校
7 . 如图,在△ABD中,AB
AD,以AB为直径的圆交AD于点M,交BD于点O,延长AO至点C,使OC
AO,连结CD,BC.
(1)求证:四边形ABCD是菱形;
(2)若AM
3,BO![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6706fe00b4e231e62d9ecbec567d526b.png)
,求cos∠DAB.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6706fe00b4e231e62d9ecbec567d526b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6706fe00b4e231e62d9ecbec567d526b.png)
(1)求证:四边形ABCD是菱形;
(2)若AM
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6706fe00b4e231e62d9ecbec567d526b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6706fe00b4e231e62d9ecbec567d526b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/26/dbc20483-39b3-439f-85f5-089a90187d1a.png?resizew=207)
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2021九年级·全国·竞赛
8 . 如图,
分别是
的三边
上任意一点,证明:
中至少有一个三角形的面积不大于
的面积的四分之一.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c5b88ec996d2d117987e7303cefe4ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b372e638a9dbed3bbf92454d816d212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0066a64af88ca17687b58a5b135ea3f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/10/627731cb-39db-4403-847b-c5ed0f1947f4.png?resizew=148)
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9 . 在直径为5的圆内放入10个点,证明其中必有两点的距离小于2.
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10 . 如图:
中,
,
,
是
的外角平分线与
的外接圆的交点.点
在
上且
,已知
,
,求
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abcb980da8e86b4cfd322616dc84fc6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f81e24376a13d648c2ed0dc73bc710e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/992a19339adac3a2f1aab1fbc11b1c68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0cee0f36dc452e58086832c0152b641.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08e82a0052299f414cff1e7afcc5aa19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/23/a371b911-49c3-4c0a-8fd2-afb5cb654047.png?resizew=135)
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