【问题呈现】
和
都是直角三角形,
,
,
,连接
,
,探究
,
的位置关系.
时,直接写出
,
的位置关系: __________;
(2)如图2,当
时,(1)中的结论是否成立?若成立,给出证明;若不成立,说明理由.
【拓展应用】
(3)当
,
,
时,将
绕点C旋转,使A,D,E三点恰好在同一直线上,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c41663d9e5e18891475aeaa98794f33d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d631f45bc652539853f236952afa5bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc75953bf5dcfa4af308c34bf9952d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f99100a3ec33eecf804076f76c905359.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4818e07559cd9ec3810dcb4d0cdb6a92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0086b054ef120408acac806a1b1318.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
(2)如图2,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/060e7930731eddbcfac592b808e9b698.png)
【拓展应用】
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94a71a658362bd9faa329c3d9f6e6d2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b8b709a173120436dac669c74b927d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/377b5f7197e5bd1afeea4d931307956a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d631f45bc652539853f236952afa5bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
更新时间:2024-01-23 09:24:51
|
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【推荐1】小明是魔方爱好者,他擅长玩各种魔方,从二阶魔方到九阶魔方,他都能成功复原.有一天,小明突然想到一个问题,在九阶魔方中,到底含有多少个长方体呢?为此,我们先来解决这样一个数学问题:如图,图1是一个长、宽、高分别为a,b,c(a≥2,b≥2,c≥2,且a,b,c是正整数)的长方体,被分成了a×b×c个棱长为1的小立方体.这个几何体中一共包含多少个长方体(包括正方体)?(参考公式:1+2+3…+n
).
探究一:如图2,该几何体有1个小立方体组成,显然,该几何体共有1个长方体.如图3,该几何体有2个小立方体组成,那么它一共包含1+2=3个长方体.如图4,该几何体有3个小立方体组成,那么它一共包含 个长方体.如图5,该几何体﹣共包含210个长方体,那么该几何体共有 个小立方体组成.
探究二:如图6,该几何体有4个小立方体组成,那么它一共包含(1+2)×(1+2)=9个长方体.如图7,该几何体有6个小立方体组成,那么它一共包含 个长方体.如图8,该几何体共有2m个小立方体组成,那么该几何体一共有 个长方体.
探究三:如图1,该几何体共有个a×b×c小立方体组成,那么该几何体共有 个长方体.
探究四:我们现在可以解决小明开始的问题了.在九阶魔方(即a=b=c=9)中,含有 个长方体.
探究五:聪明的小明在学习了三种视图后,又提出一个新的问题:在图1中,若a=6,b=4,c=5,如果拿走一些小立方体后,剩下几何体的三种视图与原图1的三种视图完全一样,那么最多可以拿走 个小立方体;此时,剩下的几何体的表面积是 .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4878947f03747c13d62540fde471b985.png)
探究一:如图2,该几何体有1个小立方体组成,显然,该几何体共有1个长方体.如图3,该几何体有2个小立方体组成,那么它一共包含1+2=3个长方体.如图4,该几何体有3个小立方体组成,那么它一共包含 个长方体.如图5,该几何体﹣共包含210个长方体,那么该几何体共有 个小立方体组成.
探究二:如图6,该几何体有4个小立方体组成,那么它一共包含(1+2)×(1+2)=9个长方体.如图7,该几何体有6个小立方体组成,那么它一共包含 个长方体.如图8,该几何体共有2m个小立方体组成,那么该几何体一共有 个长方体.
探究三:如图1,该几何体共有个a×b×c小立方体组成,那么该几何体共有 个长方体.
探究四:我们现在可以解决小明开始的问题了.在九阶魔方(即a=b=c=9)中,含有 个长方体.
探究五:聪明的小明在学习了三种视图后,又提出一个新的问题:在图1中,若a=6,b=4,c=5,如果拿走一些小立方体后,剩下几何体的三种视图与原图1的三种视图完全一样,那么最多可以拿走 个小立方体;此时,剩下的几何体的表面积是 .
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【推荐2】已知抛物线
.
(1)无论
取何值,抛物线恒过点________;
(2)该抛物线的顶点随
的变化而变化,请求出抛物线顶点纵坐标的最大值;
(3)已知点
、
,抛物线与线段
只有一个交点,求该抛物线顶点横坐标的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/440f304cd45c08f742449895867d619f.png)
(1)无论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)该抛物线的顶点随
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa3305ac6ef7117a61b3ed9fffb03030.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18782d85f69b5dcdbdd87667b65d8585.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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【推荐1】如图,AB为
的直径,点P是
上不与A,B重合的一个动点,延长PA到C,使
,点D为
上一点,且满足
,射线CD交PB的延长线于点E.
![](https://img.xkw.com/dksih/QBM/2020/5/19/2466009303736320/2466113809973248/STEM/563a95c82e0b43d6bdad25a60de6d4ab.png?resizew=202)
(1)求证:
;
(2)填空:
①若
,则四边形ABED的最大面积为________;
②若射线CD与
的另一个交点为F,则当
________时,以O,A,D,F为顶点的四边形为菱形.
![](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/d947a7d8dc459d19e7d5be243a122c2a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98e8974c64655d4caa8b0fc6cc8c7807.png)
![](https://img.xkw.com/dksih/QBM/2020/5/19/2466009303736320/2466113809973248/STEM/563a95c82e0b43d6bdad25a60de6d4ab.png?resizew=202)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07076fe6132133c04a38cd2288ec0608.png)
(2)填空:
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54275b7e571660d0a9e0370fbfe5050b.png)
②若射线CD与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5961764f4c33ea249a94a3a03818549d.png)
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名校
【推荐2】在
中,
,点
为
延长线上任一点,连接
.
,若
,
,求线段
的长;
(2)如图
,将线段
绕着点
逆时针旋转
得到线段
,连接
.点
为
的中点,连接
.求证:
;
(3)在(
)的条件下,设点
为直线
上的点,
交
于点
.点
在
延长线上运动的过程中,当
时,将
沿直线
翻折到
所在平面内得到
,同时将
沿直线
翻折到
所在平面内得到
.在
取得最大值时,请直接写出
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb9af00d442a5c693c970f30efcc916f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/714cc3707bba3bfdb56e251999be8592.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(2)如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40b3e9a99b5b8258baab51f66d1a48ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56a2781065de8c58734a9e8dcf30083f.png)
(3)在(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3834d7ec7531f3c3c0ce9b286f7a49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf4dc4d7d30af1cdce660795e0fd7d7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0702bd0b088224b2aedcc925f313d93f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9dcb640d3fec9c0eb651f5ece16829a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65bb1c5af4c7a9376882867e07690b18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9dcb640d3fec9c0eb651f5ece16829a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12aa2f3e1f0a873a999e236bc337e190.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72b8462b1655b13a1f0ef0e30c3ba84c.png)
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【推荐1】如图,在
中,
,
,
,点P从点B出发沿着边
以
的速度匀速向终点A运动;同时点Q从点C出发沿着边
以
的速度匀速向终点B运动,运动时间为t秒
,连结
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/2/6667e11b-9e8b-4c4b-aa37-85b23b04685b.png?resizew=146)
(1)
______
.
(2)当
与
相似时,求t的值.
(3)点M为
的中点,在点P与点Q运动的过程中,直接写出点M运动的路径长.
(4)以线段
和线段
为邻边,构造平行四边形
,连结
.直接写出直线
与
的边垂直时t的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e3262fc038bbec5e7c8cc47df08bef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dafddfd80f69337909f6eac0f0dbfd59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/432043f79e122f291c47453013042704.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dea2ae9d515f9ab351ad72306b776ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb73f3d840fcda9bf55aef0aa147e21f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b3c2d90ab6cea17fc95433e3644ee3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0deac44991b2d068632ca89a3448c2df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/2/6667e11b-9e8b-4c4b-aa37-85b23b04685b.png?resizew=146)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c26a46e7879436d532af3f4b6e258a81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8dcc9f79fe5f07f25447aa442ee14ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc3322a2ad9a95bdc9fc576a7a158d4d.png)
(3)点M为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
(4)以线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aee839b2aeb11e2ac709d78cd213db47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d567bdeba9b8e17d0911f594e141eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
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【推荐2】如图,圆E是三角形ABC的外接圆,∠BAC=45°,AO⊥BC于O,且BO=2,CO=3,分别以BC、AO所在直线建立坐标轴.
(1)求三角形ABC的外接圆直径;
(2)求过ABC三点的抛物线的解析式;
(3)设P是(2)中抛物线上的一个动点,且三角形AOP为直角三角形,则这样的点P有几个?(只需写出个数,无需解答过程).
(1)求三角形ABC的外接圆直径;
(2)求过ABC三点的抛物线的解析式;
(3)设P是(2)中抛物线上的一个动点,且三角形AOP为直角三角形,则这样的点P有几个?(只需写出个数,无需解答过程).
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/27/da40cccd-858f-4fb8-9bac-02640286ab48.png?resizew=146)
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【推荐1】已知抛物线
(a,b,c为常数,且
与x轴交于
,
两点(点
在点
的左侧).
(1)当
,
且
时,求
,
两点的坐标(可用含c的式子表示);
(2)若抛物线与y轴交于点C,当
ABC是直角三角形时,求ac的值;
(3)若抛物线与x轴只有一个公共点M(2,0),与y轴交于(0,2),直线l:
与抛物线交于P、Q两点(P在Q的左侧),过点P且与y轴平行的直线与直线MQ相交于点N,判断点N的纵坐标是否为一个定值?如果是,请求出这个定值;如果不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a90385c676848de67293e3ed6bc000fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f28b08682efa2692b052f64fe1448fce.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b550ee821ee1838384835e81fc34b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9801ac7225ea5a292c4cb92a43638d15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f75807858b7804a1ad2039c41f323a18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(2)若抛物线与y轴交于点C,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4cba95fc7d4853a243f8e3fb20ce70.png)
(3)若抛物线与x轴只有一个公共点M(2,0),与y轴交于(0,2),直线l:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d103965b7e9c5c97eb050aaa23027c0.png)
您最近一年使用:0次
【推荐2】已知,在
中,
.
![](https://img.xkw.com/dksih/QBM/2021/6/17/2745053374537728/2745112297226240/STEM/c8de4741-94b8-4782-9d89-11495dbd295e.png)
(1)如图1,已知点D在
边上,
,连结
.试探究
与
的关系;
(2)如图2,已知点D在
下方,
,连结
.若
,
,
,
交
于点F,求
的长;
(3)如图3,已知点D在
下方,连结
、
、
.若
,
,
,
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40975f7553d8cfa57951b568bae9c464.png)
![](https://img.xkw.com/dksih/QBM/2021/6/17/2745053374537728/2745112297226240/STEM/c8de4741-94b8-4782-9d89-11495dbd295e.png)
(1)如图1,已知点D在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82e4d16711dcb484b083d8a5d54f8df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
(2)如图2,已知点D在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82e4d16711dcb484b083d8a5d54f8df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93380ab8eccc6e6f11bdf19ffc2d4cd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80b1107f50f4fc687e2cc5a11fb4486e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0761165f1176f3a5fe4f7b052832316d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
(3)如图3,已知点D在
![](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/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45723dce96b74f027adcb1a4ecd19db8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc61db5980a1eeedcedaa277bbec981c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cee80471cdc55b089acf7a446ba1157.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/988fafe3eeef045a7e6c2d2c3cfe3ef5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e61b8262aebd3e6858cc051af3db162f.png)
您最近一年使用:0次