【教材呈现】如图是华师版九年级上册数学教材第103-104页的部分内容.
【定理证明】(1)小明根据教材图2的提示,证明过程为:延长
至点
,使
,连接
、
…,结合图①帮助小明完成直角三角形的性质:“直角三角形斜边上的中线等于斜边的一半”的证明.
【定理应用】(2)如图②,在
中,
,垂足为点
,
是
边的中线,
垂直平分
,则
,则
为________度.
【拓展提高】(3)如图③,在
中,
,
,
恰好是中线,则
的度数为________度.
如图1,画![]() ![]() ![]() ![]() ![]() ![]() ![]() 已知:如图2,在 ![]() ![]() ![]() 求证: ![]() ![]() |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0fa1aa5a7a5bb172ed4603f17c8b2c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
【定理应用】(2)如图②,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b5f215a42c4b7078d8d65923eb9980e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e72b2e1ff83e95df048745322982451.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a98f404d1abc3c8b53a93681b2f1636.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db866ecc0c098bfc32ec8a43b65bdd58.png)
【拓展提高】(3)如图③,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d3c9ce32b721995f355eea411340e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a22b02f6c8636152aeb62d191251e931.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/194741f4d2ae7ee44cafca780361446a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/3/7cf856e9-ac47-4041-9dd2-2bb0ed051e3e.jpg?resizew=308)
更新时间:2024-03-03 21:07:55
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解答题-作图题
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【推荐2】如图,在等边
中,点D在BA的延长线上,点P是BC边上的一个动点(点P不与点B重合),将线段PD绕点P逆时针旋转60°得到线段PE,连接BE和DE.
![](https://img.xkw.com/dksih/QBM/2022/5/30/2990578452447232/2991230717550592/STEM/63427d4f-798c-45f2-a63d-1ebf1a597f33.png?resizew=135)
(1)依据题意,补全图形;
(2)比较
与
的大小,并证明;
(3)用等式表示线段BE、BP与BD之间的数量关系,并证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://img.xkw.com/dksih/QBM/2022/5/30/2990578452447232/2991230717550592/STEM/63427d4f-798c-45f2-a63d-1ebf1a597f33.png?resizew=135)
(1)依据题意,补全图形;
(2)比较
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d3e081637bcea5368cc72370bae283b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ffab06abe01c9b48aa4692c73d8b7c6.png)
(3)用等式表示线段BE、BP与BD之间的数量关系,并证明.
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【推荐1】如图,AB是半圆O的直径,AC是弦,点P从点B开始沿BA边向终点A以1cm/s的速度移动,若AB长为10cm,点O到AC的距离为4cm.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/6/c9cd655b-64e8-4f43-a1d5-43d2f4548417.png?resizew=194)
(1)求弦AC的长;
(2)求△APC是等腰三角形时,点P运动的时间.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/6/c9cd655b-64e8-4f43-a1d5-43d2f4548417.png?resizew=194)
(1)求弦AC的长;
(2)求△APC是等腰三角形时,点P运动的时间.
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【推荐2】如图,在
中,
,D为BC的中点,
,垂足为点E,
,交CE的延长线于点F.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/13/cbffa695-0d87-49d0-bac8-43ac79bec7fe.png?resizew=194)
(1)求证:
;
(2)求证:AB垂直平分DF.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e64f5b5c0dd360f325604eaf7863bd7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/497846628a41a9bc750a645e045afb47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f572de8798c6eb993ede7606cf7402e9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/13/cbffa695-0d87-49d0-bac8-43ac79bec7fe.png?resizew=194)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4a35c74cc181f1693e7720a3a562d1c.png)
(2)求证:AB垂直平分DF.
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【推荐1】【教材呈现】下图是华师版九年级上册数学教材102﹣103页的部分内容.
性质:直角三角形的斜边中线等于斜边的一半给出上述性质证明中的部分演绎推理的过程如下:已知:如图1,在△ABC中,∠ACB=90°,CD为斜边AB上的中线.求证:CD=
AB证明:如图2,延长CD至点E,使DE=CD,连接AE,BE.
![](https://img.xkw.com/dksih/QBM/2022/2/13/2915426579021824/2924622911520768/STEM/9914c3009099450ea9dbd396115ba57d.png?resizew=496)
【问题解决】请结合图3将证明过程补充完整.
【应用探究】
(1)如图4,在△ABC中,AD是高,CE是中线,点F是CE的中点,DF⊥CE,点F为垂足,∠AEC=78°,则∠BCE为 度.
(2)如图5,在线段AC上有一点B,AB=4,AC=11,分别以AB和BC为边作正方形ABED和正方形BCFG,点E落在边BG上,连接DF,点H为DF的中点,连接GH,则GH的长为 .
性质:直角三角形的斜边中线等于斜边的一半给出上述性质证明中的部分演绎推理的过程如下:已知:如图1,在△ABC中,∠ACB=90°,CD为斜边AB上的中线.求证:CD=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e244c12208e7c6a01e3580d1ae9d7544.png)
![](https://img.xkw.com/dksih/QBM/2022/2/13/2915426579021824/2924622911520768/STEM/9914c3009099450ea9dbd396115ba57d.png?resizew=496)
【问题解决】请结合图3将证明过程补充完整.
【应用探究】
(1)如图4,在△ABC中,AD是高,CE是中线,点F是CE的中点,DF⊥CE,点F为垂足,∠AEC=78°,则∠BCE为 度.
(2)如图5,在线段AC上有一点B,AB=4,AC=11,分别以AB和BC为边作正方形ABED和正方形BCFG,点E落在边BG上,连接DF,点H为DF的中点,连接GH,则GH的长为 .
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【推荐2】如图,已知
中,
,
.
![](https://img.xkw.com/dksih/QBM/2022/5/11/2976850215952384/2977837281484800/STEM/3b6b83dc77594864b97cf33530347485.png?resizew=205)
(1)请判断
的形状,并说明理由.
(2)点D为AB边上一点,且
,
①求
的度数.
②当
时,求CD的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b10134e7a46e6f6f7cb9d5e2371727d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dd1443ee6c8c697f812381f5153a796.png)
![](https://img.xkw.com/dksih/QBM/2022/5/11/2976850215952384/2977837281484800/STEM/3b6b83dc77594864b97cf33530347485.png?resizew=205)
(1)请判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(2)点D为AB边上一点,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da4e859c1bddc350b4a3df1b4e390aa1.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdb8eca20ce2c918ea4034ea15210c7f.png)
②当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305a88d4e0249bd16d48eda01331d2d4.png)
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【推荐1】如图1,为放置在水平桌面
上的台灯,底座的高
为
.长度均为
的连杆
,
与
始终在同一水平面上.
![](https://img.xkw.com/dksih/QBM/2019/6/24/2232498479439872/2234525091020800/STEM/de0d9f1b7d1c47b49eee02898039b3a8.png?resizew=470)
(1)旋转连杆
,
,使
成平角,
,如图2,求连杆端点
离桌面
的高度
.
(2)将(1)中的连杆
绕点
逆时针旋转,使
,如图3,问此时连杆端点
离桌面
的高度是增加了还是减少?增加或减少了多少?(精确到
,参考数据:
,
)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e62a0233031d4e5ecc0b60b51ac25ecb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0731be9fc9a06a97db4857ccd2240772.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/2019/6/24/2232498479439872/2234525091020800/STEM/de0d9f1b7d1c47b49eee02898039b3a8.png?resizew=470)
(1)旋转连杆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/727ad3e630a224303d6d3b8ad5c114ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69b755ee4dda4a72063c6fba125720a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
(2)将(1)中的连杆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36eaa25934be72ea6afa4de50d60951a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/263a592442ba4c97ad21d02216e720a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c894b7d6baa55c80c64e74748dad898.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/460317e7c26f95b9b29cfe1a89b796d6.png)
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【推荐2】如图,在矩形
中,对角线
和
相交于点
,
,
的平分线交
于点
,连接
,求
的度数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc1eb76fe74cba30f7cbcde349ba80da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3981e7286d41960daf4e110c1c84e03a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a299d2b999568e80be8005565ba209a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ea2f6098be91fa704dc562b62471362.png)
![](https://img.xkw.com/dksih/QBM/2018/5/10/1942240160800768/1943785195667456/STEM/c638efc9-169a-4373-933f-53f9177f3339.png?resizew=187)
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