如图,以菱形ABCD对角线交点为坐标原点,建立平面直角坐标系,A、B两点的坐标分别为(﹣2
,0)、(0,﹣
),直线DE⊥DC交AC于E,动点P从点A出发,以每秒2个单位的速度沿着A→D→C的路线向终点C匀速运动,设△PDE的面积为S(S≠0),点P的运动时间为t秒.
![](https://img.xkw.com/dksih/QBM/2019/6/6/2219620991893504/2221870286561280/STEM/0008954218174b53ab47b12ee1e52394.png?resizew=207)
(1)求直线DE的解析式;
(2)求S与t之间的函数关系式,并写出自变量t的取值范围;
(3)当t为何值时,∠EPD+∠DCB=90°?并求出此时直线BP与直线AC所夹锐角的正切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
![](https://img.xkw.com/dksih/QBM/2019/6/6/2219620991893504/2221870286561280/STEM/0008954218174b53ab47b12ee1e52394.png?resizew=207)
(1)求直线DE的解析式;
(2)求S与t之间的函数关系式,并写出自变量t的取值范围;
(3)当t为何值时,∠EPD+∠DCB=90°?并求出此时直线BP与直线AC所夹锐角的正切值.
更新时间:2019-06-09 13:49:34
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解答题-证明题
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【推荐1】在菱形
中,
,
,动点
从
点出发沿线段
以每秒钟1个单位的速度向
点运动(到达
点后停止),设点
运动时间为
,将点
绕着点
顺时针旋转
,得到对应点
,连接
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/12/29a74040-5b53-4a30-beee-c3b6150b7bdb.png?resizew=514)
(1)如图1,求证:
;
(2)如图2,连接
,若
与△
相似,求
的值;
(3)若点
关于直线
的对称点
在菱形的边上,请直接写出
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7788830ed1cb3b9c5988f70f43595f2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c26ed4600cef25f040caa553ac21051.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2684b10100be43f77a13fa0ccd1c1d72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a89638d4a4e6dc787ec2e3461c273fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c884b508394b3ab50734b584d9ec783c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/12/29a74040-5b53-4a30-beee-c3b6150b7bdb.png?resizew=514)
(1)如图1,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bceef8a3f331812680b1590182a827f4.png)
(2)如图2,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ada5b725eb3be036a2dd999bd613307.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f216bdf0ff1340c1209c6503e1946d5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(3)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42198b8e646ccfbbec323fd364c4d90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39e210795a966bd5dc1293479ab1988b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
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【推荐2】如图1,在□ABCD中,
,
,
,射线AE平分
动点P以
的速度沿AD向终点D运动,过点P作
交AE于点Q,过点P作
,过点Q作
,交PM于点
设点P的运动时间为
,四边形APMQ与四边形ABCD重叠部分面积为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/349edc17cbf6bb3ff82cdcff68364454.png)
![](https://img.xkw.com/dksih/QBM/2018/8/11/2008020079321088/2010137564880896/STEM/ed5d49662c634a529e6decbeaa7dce68.png?resizew=147)
______
用含t的代数式表示![](https://staticzujuan.xkw.com/quesimg/Upload/formula/987517758fad59f6f695761deb2a5ebd.png)
当点M落在CD上时,求t的值.
求S与t之间的函数关系式.
如图2,连结AM,交PQ于点G,连结AC、BD交于点H,直接写出t为何值时,GH与三角形ABD的一边平行或共线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eba53fc741181ef72e8722524d0fc613.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76821ee1bf83e4bed314a6c98c29633a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fdb15f442539ba60f4c69638f8c63bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1bad971098f94b950cb6d3c3e0c92a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f601546a19f1b6ae7ba6fbc5c08706d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a01fd67919faeadbc8ad4bc887c0a39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d17878c2ad3883cb0ff16cc603fdd43.png)
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【推荐1】如图,在
中,
是边
的中点,
是
上靠近点
的三等分点,连接
,延长
至点
,使得
,连接
.
(1)试判断
与
的位置关系,并说明理由;
(2)
与
交于点
,若
,求证:
是
的中点;
(3)在(2)的条件下,连接
,求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
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![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/18/c5b07ece-1ee1-49cb-973b-0f8dc4711e33.png?resizew=265)
(1)试判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f864b11496cc738c29a737267cfc07cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
(3)在(2)的条件下,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06e322e0c87479bba874db9ae9ba36b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3689c3d2b5b3df089e832954065d6493.png)
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【推荐2】
在平面直角坐标系中,点O为坐标原点,直线y=﹣x+4与x轴交于点A,过点A的抛物线y=ax2+bx与直线y=﹣x+4交于另一点B,且点B的横坐标为1.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/26/7f9729f2-c1df-4315-a4f5-fe9093d93c7f.png?resizew=389)
(1)该抛物线的解析式为;
(2)如图1,Q为抛物线上位于直线AB上方的一动点(不与B、A重合),过Q作QP⊥x轴,交x轴于P,连接AQ,M为AQ中点,连接PM,过M作MN⊥PM交直线AB于N,若点P的横坐标为t,点N的横坐标为n,求n与t的函数关系式;在此条件下,如图2,连接QN并延长,交y轴于E,连接AE,求t为何值时,MN∥AE.
(3)如图3,将直线AB绕点A顺时针旋转15度交抛物线对称轴于点C,点T为线段OA上的一动点(不与O、A重合),以点O为圆心、以OT为半径的圆弧与线段OC交于点D,以点A为圆心、以AT为半径的圆弧与线段AC交于点F,连接DF.在点T运动的过程中,四边形ODFA的面积有最大值还是有最小值?请求出该值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe3a78950f6a4bb479c7ec0d8e57b5b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/26/7f9729f2-c1df-4315-a4f5-fe9093d93c7f.png?resizew=389)
(1)该抛物线的解析式为;
(2)如图1,Q为抛物线上位于直线AB上方的一动点(不与B、A重合),过Q作QP⊥x轴,交x轴于P,连接AQ,M为AQ中点,连接PM,过M作MN⊥PM交直线AB于N,若点P的横坐标为t,点N的横坐标为n,求n与t的函数关系式;在此条件下,如图2,连接QN并延长,交y轴于E,连接AE,求t为何值时,MN∥AE.
(3)如图3,将直线AB绕点A顺时针旋转15度交抛物线对称轴于点C,点T为线段OA上的一动点(不与O、A重合),以点O为圆心、以OT为半径的圆弧与线段OC交于点D,以点A为圆心、以AT为半径的圆弧与线段AC交于点F,连接DF.在点T运动的过程中,四边形ODFA的面积有最大值还是有最小值?请求出该值.
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【推荐3】类比、转化、从特殊到一般等思想方法,在数学学习和研究中经常用到.小明在数学学习中遇到了这样一个问题:“如图1,Rt△ABC中,∠ACB=90°,∠CAB=α,点P在AB边上,过点P作PQ⊥AC于点Q,将△APQ绕点A逆时针方向旋转,如图2,连接CQ.O为BC边的中点,连接PO并延长到点M,使OM=OP,连接CM.探究在△APQ的旋转过程中,线段CM,CQ之间的数量关系和位置关系,”小明计划采用从特殊到一般的方法探究这个问题.
(1)填空:如图3,当α=30°时,
= ,直线CQ与CM所夹锐角的度数为 ;如图4,当α=45°时,
= ,直线CQ与CM所夹锐角的度数为 ;
(2)将△APQ绕点A逆时针方向旋转的过程中,线段CQ,CM之间的数量关系如何?直线CQ与CM所夹锐角的度数是多少(用含α的式子表示)?请仅就图2所示情况说明理由;
(3)如图4,在Rt△ABC中,若AB=4,α=45°,AP=3,将△APQ由初始位置绕点A逆时针方向旋转β角(0°<β<180°),当点Q到直线AC的距离为2时,请直接写出线段CM的值.![](https://img.xkw.com/dksih/QBM/2021/5/13/2720028244099072/2720082525274112/STEM/2f0c6fe4cbc84b2f8090d2ad0ae743db.png?resizew=28)
(1)填空:如图3,当α=30°时,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d899ebf02955ad10da3df78cc80b06e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d899ebf02955ad10da3df78cc80b06e2.png)
(2)将△APQ绕点A逆时针方向旋转的过程中,线段CQ,CM之间的数量关系如何?直线CQ与CM所夹锐角的度数是多少(用含α的式子表示)?请仅就图2所示情况说明理由;
(3)如图4,在Rt△ABC中,若AB=4,α=45°,AP=3,将△APQ由初始位置绕点A逆时针方向旋转β角(0°<β<180°),当点Q到直线AC的距离为2时,请直接写出线段CM的值.
![](https://img.xkw.com/dksih/QBM/2021/5/13/2720028244099072/2720082525274112/STEM/2f0c6fe4cbc84b2f8090d2ad0ae743db.png?resizew=28)
![](https://img.xkw.com/dksih/QBM/2021/5/13/2720028244099072/2720082525274112/STEM/a930cc46-82ec-4063-9747-831919686a70.png)
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