1 . 如图,
中,
,以它的各边为边向外作三个正方形,面积分别为
、
、
已知
,
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef354e5c5ff828cc8d27c71badd40f98.png)
______ .
![](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/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd8afa8fea9c868eacb6a4f35362a30d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc19cad4e47fb1dfdfc6cc912c49d664.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef354e5c5ff828cc8d27c71badd40f98.png)
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2 . 勾股定理是数学史上的两个宝藏之一,小亮在学习完本章知识后,他和星源数学社的其他成员进行了有关知识的探索.请你根据他们的思路完成下列各项内容:
问题解决:如下图
中,
,分别以其三边向形外作正方形,若
,
,则
______.
(1)如下图,若以
的三边向形外作等腰直角三角形,
,
,则
、
、
之间的关系为______.
、
、
之间的关系为______.
中,
,分别以它的三边向形外作平行四边形,
交
于P,交
于N,且
,若平行四边形
和平行四边形
的面积分别为10和8,则平行四边形
的面积为______.
问题解决:如下图
![](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/6e1c6c69a4411995616551f18eb0d49b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4582b6ba00094ea27334db8284957a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae248960d8c1677cf948f8251275e863.png)
(1)如下图,若以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/631768f5428cb8bd9d79a0ef2832da72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b80f6fceef16ab9ad88ecd902d9c86b9.png)
![](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/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](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/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](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/dc8f51951ccccbf1e912747fda23e432.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42887d9bf31c1dd99f13c39e63c9ab9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c89090a512721b3da471cdabedeefc8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e835b13c9aae88df4607e0c7f4db708.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/736df44e05c14932d3dfb3b32ebd2bfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de6d1f7b9ec2c9edace6a4c7569bf040.png)
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3 . 勾股定理的证明:
如图1,在
中,
.求证:
.
(1)看到要证明的结论,想到小学学习的正方形的面积计算方法是
,受此启发,要证明
,于是分别以
的三边
、
、
为边向
的外面作正方形,如图2,只需证明_____+____
即可;
分成两个长方形,使其面积分别等于其余两个正方形的面积呢?此处遇到困难,于是查阅资料,发现欧几里得的《几何原本》中,在图2的基础上作了如图3中的辅助线,小丽尝试理解辅助线是如何想到的.
边的垂线,垂足为点M,交
于点N,就实现将正方形分成两个长方形的目的,只需证明
,
_______;
②要想建立正方形
和长方形
面积的关系,只能将其分别建立与
和
的面积关系,易得
,_____
,而
( )(填推理依据),于是
、同理将正方形
的面积转化为另一长方形的面积,小丽通过体验勾股定理的探索过程,发现利用面积证法将未知问题逐步转化为已知问题.
如图1,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/f4371a4e0c848f28f263b10c1ee6b61a.png)
(1)看到要证明的结论,想到小学学习的正方形的面积计算方法是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8596dec4570b5ff4668bdeae6afd4ccb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4371a4e0c848f28f263b10c1ee6b61a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.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/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7eeaf78042a213f55ecd1ab33fb11023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a34fdf9e6d2d87d01ad0bbb6a73ee05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63e36329f5e0979f5ee776ac5d06327.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dc91c2e47d6ccdf1d42c473c6796f3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/675c6bea2366d9cefdb7f912e9a08a0c.png)
②要想建立正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b9d54cbbf601f4583659771eb534997.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9847ed8842ddadce59c2b556f8fa5709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb1975abdfba3d89ee75a40e4239f6e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce8444310731eab0b267e2c2aba45119.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9957384cd5fdf27164484b9452b373d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc4b4c01c0513f4f4338a89792f186b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dc91c2e47d6ccdf1d42c473c6796f3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5f5e620dccc65cf6138f6d67a3e057b.png)
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4 . 如图,在
中,
,分别以
,
为边向外作正方形,面积分别为
,
,若
,
,则
的长为( )
![](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/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](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/71cbe3cea00e2e1b5b2283c7626ffc93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4582b6ba00094ea27334db8284957a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
A.4 | B.2 | C.![]() | D.3 |
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2024-06-02更新
|
413次组卷
|
2卷引用:陕西省西安市阎良区2023-2024学年八年级下学期期中数学试题
5 . 如图,在
中,
,以
的三边为边向外作正方形,其面积分别为
,
,
,
,
,则
( )
![](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/dd967903ed5a6f640a5b801ec8be0070.png)
![](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/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/053788861aab1c486f8a954efdf16200.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71148c104abc89a4fdf81909917dd62a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1b625192d7398634a02ea24fa78ed87.png)
A.9 | B.12 | C.15 | D.144 |
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6 . 如图,在直线上依次摆放着7个正方形,已知斜放置的3个正方形的面积分别是
,
,
,正放置的4个正方形的面积依次是
,
,
,
,则
的值为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](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/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f30f56664446f32dbbc2c5f12a99374.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/838a984dff7abdc750c97afd75e962eb.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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7 . 如图,由一个直角三角形和三个正方形组成的图形,其中最右边的正方形边长为3,则阴影部分面积是______ .
您最近一年使用:0次
名校
8 . 正方形
的边长为1,其面积记为
,以
为斜边作等腰直角三角形,以该等腰直角三角形的一条直角边为边向外作正方形,其面积记为
,…按此规律继续下去,则
的值为________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb77e7e99fc7b752b317cde8ac2fbb4c.png)
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名校
9 . 如图,正方形
的边长分别为直角三角形的三边长,若正方形
的边长分别为4和8,则正方形
的面积为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
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10 . 如图,网格中每个小正方形的面积为单位1,则图形C的面积是( )
A.6 | B.![]() | C.![]() | D.13 |
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