上册第二章第1-5节勾股定理;神秘的数组;平方根;立方根;实数

第二章第1-5节勾股定理;神秘的数组;平方根;立方根;实数 年 级 初二 学 科 数学 版 本 江苏科技版 内容标题 勾股定理与平方根 编稿老师 郭红 【本讲教育信息】 一. 教学内容: 勾股定理与平方根 勾股定理及逆定理揭示了形和数之间的紧密联系，在现实生活中也有着广泛的应用，体现了数学的价值。而且，在数学发展史上有着重要的地位，对人类的发展也起着重要的作用。 平方根、算术平方根、立方根概念的引入，体现了引入新数的必要性。从而把对数的认识上升到“实数”上。 二. 重点、难点: 1. 勾股定理及逆定理的理解与应用。 2. 无理数和实数的概念，实数与数轴上的点一一对应。 3. 近似数字与有效数字的概念。 三. 知识要点: . 勾股定理:直角三角形两直角边的平方和等于斜边的平方 1 222a、b、ca，b,c 2. 勾股定理逆定理:如果一个三角形的三边满足，那么这个三角形是直角三角形。(判定三角形是直角三角形的一种) 222a、b、ca，b,c 满足的三个正整数称为勾股数。如:3、4、5;5、12、13等 3. 平方根与算术平方根 (1)平方根 如果一个数的平方等于a，那么这个数叫做a的平方根，也称二次方根。 2x,a即:若，那么x就叫做a的平方根。 ,a用示，读作:“正、负根号a” 22如: 3,9,(,3),9,所以，3、,3都是9的平方根。 ,2,注意,一个正数有两个平方根，它们互为相反数;如:2的平方根为 0,0 0只有一个平方根，它是0本身;即 负数没有平方根。 (2)算术平方根 aa正数有两个平方根，其中正的平方根，也叫做的算术平方根。 ,1515 如:15的平方根是;算术平方根是 (3)平方与开平方 求一个数的平方根的运算，叫做开平方。 technical department, quality Department, laboratory, measurement and construction team nt implementation of the "quality first, prevention first", a self check and cross-check, check, so that "anyone who was responsible for the construction." 14.2.5 material procurement and management 1, purchase plans according to the construction plan requirements. QC Department to participate in acceptance of materials, negative and seal the incorrect material. 2, according to tender documents and standards for test and evaluation of raw materials, qualified materials classified and piled up and set up a sign detailing acceptance of varieties, quantities, date and signature of management. 14.2.6 nonconforming product control and management of nonconforming products should be controlled on the spot, by construction workers, technicians and QC field fill out the unqualified report, nonconforming should withdraw from the scene and striking identity to avoid misuse. Face treated by supervision engineer's permission to continue construction. 14.2.7 check site laboratory quality supervision and inspection and test management program led by the QC, the test methods, procedures, and strictly according to the contract or a related specification, mainly raw materials and concrete scope of 平方的结果是唯一的。 在开平方的运算中，被开方数必须是非负数，开平方的结果不一定是唯一的。 4. 立方根 如果一个数的立方等于a，那么这个数叫做a的立方根，也称三次方根。 3即:若，那么x就叫做a的立方根。 x,a 3a用表示，读作:“三次根号a”。 33如:2,8,所以2是8的立方根;(,2),,8，所以,2是,8的立方根。 ,注意,正数的立方根是正数; 3 0的立方根是0;即0,0 负数的立方根是负数。 求一个数的立方根的运算，叫做开立方。 5. 实数 3(1)无理数:无限不循环小数，如: 5、7、, (2)实数:有理数和无理数统称为实数。 a)分类 b)实数和数轴 每一个实数都可以用数轴上的点来表示;反过来，数轴上每一个点都表示一个实数。实数与数轴上的点是一一对应的。 6. 近似数与有效数字 (1)由于实际中常常不需要用精确的数描述一个量，甚至在更多情况下不可能得到精确的数，用以描述所研究的量，这样的数就叫近似数。 如:某人的体重约为62公斤，这里的62就是近似数。 (2)一个近似数精确到哪一位，从左边第一个不是0的数算起，到这一位数字止，都叫做这个数的有效数字。 如:3.1415926„„3.14有三个有效数字3，1，4 , 43.5，10又如:34880(精确到千位) , (3)科学记数法 n把一个数记为 a，10(其中1,a,10,n是整数)的形式，就叫做科学记数法。 5，10如:696000记为6.96 (4)取近似值的方法——四舍五入法 指定舍去一个数的某一数位后边的数时，如果舍去的数里最左边的一位数字是5，6，7，8，9，就在留下的数字里被指定的数位上+1;如果舍去的数里最左边的一位数字是0，1，2，3，4，留下的数不变，舍去整数时，要用0替代舍去的每一个数字。 如:用四舍五入法截取462.3845到百分位，得到近似数462.38;到十分位，得近似数 2，10462.4;到十位，得4.6 technical department, quality Department, laboratory, measurement and construction team nt implementation of the "quality first, prevention first", a self check and cross-check, check, so that "anyone who was responsible for the construction." 14.2.5 material procurement and management 1, purchase plans according to the construction plan requirements. QC Department to participate in acceptance of materials, negative and seal the incorrect material. 2, according to tender documents and standards for test and evaluation of raw materials, qualified materials classified and piled up and set up a sign detailing acceptance of varieties, quantities, date and signature of management. 14.2.6 nonconforming product control and management of nonconforming products should be controlled on the spot, by construction workers, technicians and QC field fill out the unqualified report, nonconforming should withdraw from the scene and striking identity to avoid misuse. Face treated by supervision engineer's permission to continue construction. 14.2.7 check site laboratory quality supervision and inspection and test management program led by the QC, the test methods, procedures, and strictly according to the contract or a related specification, mainly raw materials and concrete scope of 【典型例题】 ,ABC例1. 已知等边三角形的边长是6cm。求: (1)高的长; AD (2)求。 ,ABC的面积S,ABC A B D C Rt,ABD解:(1)在中， 2222? AD,AB,BD,6,3,27,33cm 112(2) S,BC,AD,，6，33,93cm,ABC22233cm93cm答:高AD的长为，为 ,ABC的面积S,ABC 例2. 已知:如图，?ABD中?B=90?，?D=15?，C是BD上一点，AC=CD=8cm， 则AB= cm，BC= cm。 A B C D AC,CD,8cm,ACD解:??是等腰三角形 o，CAD,，D,15? oo，ACB,，CAD，，D,2，15,30? 12222Rt,ABCBC,AC,AB,8,4,43cm?中，， AB,AC,4cm2 说明:三角形的外角等于两个不相邻的内角和。 411.5,20.293x,2.0291,2x,202.9例3. 若，求(1) (2)中的x 411.5,20.294.115,2.029解:(1)? ? 3x,2.029又? ?3x = 4.115 x?1.372 41152029..,411502029,.(2)? ? 122029,,x.又? ?1,2 x = 41150 x?,20574.5 例4. 已知一个正方体的棱长是4厘米，再做一个正方体，使它的体积是原正方体体积的 3倍，求所做正方体棱长(精确到0.1cm) 解:设所做正方体棱长为xcm 333x,4，3x,192则 ? x?5.8cm technical department, quality Department, laboratory, measurement and construction team nt implementation of the "quality first, prevention first", a self check and cross-check, check, so that "anyone who was responsible for the construction." 14.2.5 material procurement and management 1, purchase plans according to the construction plan requirements. QC Department to participate in acceptance of materials, negative and seal the incorrect material. 2, according to tender documents and standards for test and evaluation of raw materials, qualified materials classified and piled up and set up a sign detailing acceptance of varieties, quantities, date and signature of management. 14.2.6 nonconforming product control and management of nonconforming products should be controlled on the spot, by construction workers, technicians and QC field fill out the unqualified report, nonconforming should withdraw from the scene and striking identity to avoid misuse. Face treated by supervision engineer's permission to continue construction. 14.2.7 check site laboratory quality supervision and inspection and test management program led by the QC, the test methods, procedures, and strictly according to the contract or a related specification, mainly raw materials and concrete scope of 答:所做正方体棱长为5.8cm 说明:按照近似计算的要求，在解题过程中应比要求的精确度多保留一位小数，最后结果再四舍五入到要求的形式。 例5. 用科学记数法表示91800000，正确的是( ) 5657(A)918× (B)91.8× (C)9.18× (D)9.18× 10101010 解:选D 1,a,10说明:紧扣科学记数法的定义，其中。 4例6. (1)一个数用科学记数法记为6×，这个数原来怎么记,它是几位整数, 10 4(2)一个数用科学记数法记为6.09×，这个数原来怎么记,它是几位整数, 10 4(3)一个数用科学记数法记为6.00009×，这个数原来怎么记,它有几位整数, 10 答:(1)60000，它是五位整数; (2)60900，它是五位整数; (3)60000.9，它有五位整数 例7. 如图，为了求出湖两岸的A、B两点之间的距离，一个观测者在点C设桩，使三角形ABC恰好为直角三角形。通过测量，得到AC长160米，BC长128米。问从点A穿过湖到点B有多远, 解:在直角三角形ABC中，AC,160，BC,128， 根据勾股定理可得 2222AB,AC,BC,160,128,96(米) 答:从点A穿过湖到点B有96米。 例8. 印度一作者所写的莲花问题，是用诗文形式写成的:平平湖水清可见，面上半尺生红莲;出泥不染亭亭立，忽被强风吹一边;渔人观看忙向前，花离原位二尺远;能算诸君请解题，湖水如何见深浅。 technical department, quality Department, laboratory, measurement and construction team nt implementation of the "quality first, prevention first", a self check and cross-check, check, so that "anyone who was responsible for the construction." 14.2.5 material procurement and management 1, purchase plans according to the construction plan requirements. QC Department to participate in acceptance of materials, negative and seal the incorrect material. 2, according to tender documents and standards for test and evaluation of raw materials, qualified materials classified and piled up and set up a sign detailing acceptance of varieties, quantities, date and signature of management. 14.2.6 nonconforming product control and management of nonconforming products should be controlled on the spot, by construction workers, technicians and QC field fill out the unqualified report, nonconforming should withdraw from the scene and striking identity to avoid misuse. Face treated by supervision engineer's permission to continue construction. 14.2.7 check site laboratory quality supervision and inspection and test management program led by the QC, the test methods, procedures, and strictly according to the contract or a related specification, mainly raw materials and concrete scope of 1解:如图，， CD,,BC,2,设AC,x2222Rt,ABC中，AB,AC，BC由已知，在 1又 AB,AD,x，2 1222? (x，),x，22 15解得 x,4 15答:湖水深尺。 4 【模拟】(答题时间:30分钟) 1. (1)已知在Rt?ABC中?C=90?。 ?若a=6，c=10，则b=______; ?若a=40，b=9，则c=______。 (2)已知在Rt?ABC中，?C=90?，AB=10。 ?若?A=30?，则BC=______，AC=______; ?若?A=45?，则BC=______，AC=______。 2. 若等腰三角形两边长分别为4和6，则底边上的高等于( ) 777414242 (A) (B)或 (C) (D)或 3. 一个长为10米的梯子斜靠在墙上，梯子的顶端距离地面的垂直距离为6米，如梯子顶部下滑1米，那么梯子的底端滑动_____________________米, 4. 用科学记数法记以下的数: (1)第五次人口普查时，中国人口约为1 300 000 000人，记为______; (2)太阳的半径约为696 000 000 米，记为___________; (3)光的速度约为300 000 000米/秒，记为_____________; (4)人体中约有2 500 000 000 000个红细胞，记为_____________; (5)我国的陆地面积居世界第三位，约为9 597 000平方千米，记为_______________。 317323054812,,.,.,.则5. 已知的值是( ) (A)1.096 (B)0.1096 (C)0.346 (D)3.46 33032906903..,3873382..,6. 已知 3332900000000387和,. 求的值。 7. 某人从A处沿直线到达B处，在图中，每个小方格的边长为1km，试求该人通过的路程。 technical department, quality Department, laboratory, measurement and construction team nt implementation of the "quality first, prevention first", a self check and cross-check, check, so that "anyone who was responsible for the construction." 14.2.5 material procurement and management 1, purchase plans according to the construction plan requirements. QC Department to participate in acceptance of materials, negative and seal the incorrect material. 2, according to tender documents and standards for test and evaluation of raw materials, qualified materials classified and piled up and set up a sign detailing acceptance of varieties, quantities, date and signature of management. 14.2.6 nonconforming product control and management of nonconforming products should be controlled on the spot, by construction workers, technicians and QC field fill out the unqualified report, nonconforming should withdraw from the scene and striking identity to avoid misuse. Face treated by supervision engineer's permission to continue construction. 14.2.7 check site laboratory quality supervision and inspection and test management program led by the QC, the test methods, procedures, and strictly according to the contract or a related specification, mainly raw materials and concrete scope of 8. 一个正常人的平均心跳速率约为70次/分，一年大约跳几次, 9. 在下列长度的四组线段中，不能组成直角三角形的是( ) 33 (A)3，3，3 (B)+1，,,，2 22 (C)8，15，17 (D)3.5，4.5，5.5 10. 感受勾股定理的美 technical department, quality Department, laboratory, measurement and construction team nt implementation of the "quality first, prevention first", a self check and cross-check, check, so that "anyone who was responsible for the construction." 14.2.5 material procurement and management 1, purchase plans according to the construction plan requirements. QC Department to participate in acceptance of materials, negative and seal the incorrect material. 2, according to tender documents and standards for test and evaluation of raw materials, qualified materials classified and piled up and set up a sign detailing acceptance of varieties, quantities, date and signature of management. 14.2.6 nonconforming product control and management of nonconforming products should be controlled on the spot, by construction workers, technicians and QC field fill out the unqualified report, nonconforming should withdraw from the scene and striking identity to avoid misuse. Face treated by supervision engineer's permission to continue construction. 14.2.7 check site laboratory quality supervision and inspection and test management program led by the QC, the test methods, procedures, and strictly according to the contract or a related specification, mainly raw materials and concrete scope of 【试题答案】 31.(1)?8 ?41 (2)?5，5 ?5，5 22 72. D 当腰为4，底为6时，高为;当腰为6，底为4时，高为 42 3. 解:根据勾股定理，开始的时候梯子底端距离墙面8米 当下滑1米 ，则高为5米，斜边为10米。 所以底面为75,5353,8，所以底端滑动米 8984. (1)人 (2)米 (3)3×10米/秒 1.3，106.96，10 126(4)个 (5)平方千米 2.5，109.597，10 3965. A. 1.2,4，0.3,2,2,,1.096 103030 33032906903..,3290006903,.6. 解:? ? 333873382..,00000387003382.., ? ? 3,,,00000387003382.. 227. 解: AB,4，8,45km 78. 解:约为次 365，24，60，70,3679200,3.6792，109. D 10. 略 technical department, quality Department, laboratory, measurement and construction team nt implementation of the "quality first, prevention first", a self check and cross-check, check, so that "anyone who was responsible for the construction." 14.2.5 material procurement and management 1, purchase plans according to the construction plan requirements. QC Department to participate in acceptance of materials, negative and seal the incorrect material. 2, according to tender documents and standards for test and evaluation of raw materials, qualified materials classified and piled up and set up a sign detailing acceptance of varieties, quantities, date and signature of management. 14.2.6 nonconforming product control and management of nonconforming products should be controlled on the spot, by construction workers, technicians and QC field fill out the unqualified report, nonconforming should withdraw from the scene and striking identity to avoid misuse. Face treated by supervision engineer's permission to continue construction. 14.2.7 check site laboratory quality supervision and inspection and test management program led by the QC, the test methods, procedures, and strictly according to the contract or a related specification, mainly raw materials and concrete scope of