解题方法
1 . [方法储备]如图1,在
中,
为
的中线,若
,
,求
的取值范围.中线倍长法:如图2,延长
至点
,使得
,连结
,可证明,由全等得到
,从而在
中,根据三角形三边关系可以确定
的范围,进一步即可求得
的范围.
在上述过程中,证明
的依据是______,
的范围为______;
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/6/105e9b4f-410f-4b19-9cc1-b739f8a845a4.png?resizew=452)
[思考探究]如图3,在
中,
,
为
中点,
、
分别为
、
上的点,连结
、
、
,
,若
,
,求
的长;
[拓展延伸]如图4,
为线段
上一点,
,分别以
、
为斜边向上作等腰
和等腰
,
为
中点,连结
,
,
.
①求证:
为等腰直角三角形;
②若将图4中的等腰
绕点
转至图5的位置(
,
,
不在同一条直线上),连结
,
为
中点,且
,
在
同侧,连结
,
.若
,
,求
和
的面积之差.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb5b12692517a39c320f99a479eb055.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65a3e478bb87d094e3a0af30dd10ae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b47d30006e2463739e4f7de4bc7f40fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fca905896e0786da8c85df0aa294fe3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661ff55b5ebbadfb600989af3cfce2fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
在上述过程中,证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cf5c3b03a2fde56e589166b110103dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/6/105e9b4f-410f-4b19-9cc1-b739f8a845a4.png?resizew=452)
[思考探究]如图3,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed10df4140819d5451773a45de66201b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be2e2c0d4ac2bd79f6cea7a9b1a50662.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce6c0e9de83f2e64ae33609fc08459d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27e887de7300ee5cb4d40ddfcac9a686.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adc90c2d45477e166b02359525f40aa6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d27bd71d79cb19eb554175e4ef0867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
[拓展延伸]如图4,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e95be88bc79b3c7e72d349c4cb3c88d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10ce5d6ca24e249f7172e51c154509b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d288a8ece64aaa06264a23a7c17d3c8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a424b50eaeafa6f302ffd95476cb86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b46c607b3deac746c0ef3389ad8f65c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9825e881728e22d0b8d4dab50854c433.png)
②若将图4中的等腰
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d288a8ece64aaa06264a23a7c17d3c8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.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/15a424b50eaeafa6f302ffd95476cb86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b46c607b3deac746c0ef3389ad8f65c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecf7679c8b4b1e442ce4286d4b0e9c32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fbd8e8c759858fb3d3132605d44e865.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50244defdb202cc420e1d6a910241c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82fd5e259868f7f38577bc1f2a41e4c5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/6/a92e025d-9a44-4769-b4b6-3641a1adc5a6.png?resizew=395)
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2 . 如图,在
中,
,以
的各边为边作三个正方形,点
落在
上,若
,空白部分面积为10,则
的长为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ed8f7d3d7043d4b1eb98fc5c4e2fcd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cc29330979e64e99678423e58759779.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89779b35b7e25cc8f6b02153134de444.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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3 . 如图,
是
的高,分别以线段
为边向外作正方形,其中3个正方形的面积如图所示,则第四个正方形的面积为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/3/db39bf2f-5c7a-44f6-b17f-3d0957ce390a.png?resizew=182)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1e1e31646126c42c74c62ec95eb15d0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/3/db39bf2f-5c7a-44f6-b17f-3d0957ce390a.png?resizew=182)
A.3 | B.4 | C.5 | D.6 |
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4 . 如图,在
中,
,分别以
,
,
为边在三角形外部作正方形,若以
和
为边的正方形面积分别为5和3,则以
为边的正方形面积s的值为( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/3/a3cd5598-f2fc-416b-a7cf-b16d33df7ad9.png?resizew=157)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed10df4140819d5451773a45de66201b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/3/a3cd5598-f2fc-416b-a7cf-b16d33df7ad9.png?resizew=157)
A.64 | B.8 | C.2![]() | D.34 |
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5 . 如图,数字代表所在正方形的面积,则
所代表的正方形的边长为 _____ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/2/948d66fb-668b-4fa1-b775-5492535b691a.png?resizew=100)
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6 . 将三张半圆形纸片按如图的方式摆置,半圆的直径恰好构成一个直角三角形,若知道图中两个月牙形的面积和,则一定能求出( )
A.直角三角形的面积 | B.最大半圆形的面积 |
C.较小两个半圆形的面积和 | D.最大半圆形与直角三角形的面积和 |
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7 . 如图,已知等腰直角三角形纸板
中,
.现要从中剪出一个尽可能大的正方形,则能剪出的最大正方形的面积是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/3/5e69c376-570e-4a37-885c-a5e1650732e9.png?resizew=142)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6eeb0c8df3d8c5260768c5e407b3583a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/3/5e69c376-570e-4a37-885c-a5e1650732e9.png?resizew=142)
A.![]() | B.![]() | C.25 | D.50 |
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8 . 如图所示,正方形
和正方形
的面积分别是100和36,
,则以
为直径的半圆的面积是_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ddcd5435b39971f897210aa0b66a259.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72a5857d3f49d365e7c3b46eae3baf4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67b686b84b7a2b8e2326dbb88af5e855.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
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9 . 如图,正方形A的面积为______ .
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10 . 如图,所有的三角形都是直角三角形,所有的四边形都是正方形,图中
,
…
分别表示对应正方形的面积,则下列结论不一定正确的是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/6/853d1eeb-0c50-480c-b362-ac4c28457b73.png?resizew=177)
![](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/194592cb77de8a597d5d64e1c85c3249.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/6/853d1eeb-0c50-480c-b362-ac4c28457b73.png?resizew=177)
A.![]() | B.![]() |
C.![]() | D.![]() |
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