word完整版巧用等底等高的关系求图形的面积

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巧用等底等高的关系求图形的面积
孟桂民
一、知识基础下图请同学们观察，在我们以前的学习中，我们掌握了许多基本图形的面积计算方法，中哪些图形的面积相等？为什么？ll//已知（单位：厘米）21
S??SS从图中我们看出ABE?ABD?ABC?SS?FGJKHGFI平行四边形平行四边形HGFI及△ABC和△ABDABE等底等高所以它们的面积相等，同理平行四边形因为△和平行四边形FGJK等底等高所以它们的面积也相等。

等底等底，所以平行四边形的面积是三角形面积的ABCFGHI和三角形而平行四边形2倍。

如果两个或几个三角形底相等，高也相等，从这个练习我们得到：那么这几个三角形的面积也相等。

下面还有一组图形请同学们观察，下面几个三角形面积之间有什么关系？为什么？（单位：厘米）
因为这两个三角形等底的三角形面积相等，2的三角形面积和图1从观察我们看出①图
ht a change, and eightogical security barrier, and long East energy chemical base constructi on, major strategy intensive introduced, and provi ncial eig col oriented see, S6 poverty st orming, a nd 873 traffic breakthrough, and 363 water guara ntees, major a ction speed up implementation, eopportunities for us to a nt of olhe developmed revolutionary a nd a series of policy measures in areas such as infrastructure, public services, promoting high standards, support, and more ccelerate development, provide greater support. From province s policy cisi velopment meaplanning, ncia ss of wellalleviation crucial further development to accelerate the proce-off society ha s created a great opport unity. 18
si nce the Ce ntral and provil poverty alleviation a nd development into the overall strategic priority and
irst proje瑣, formulated a series of policies now focus, de sures have all been clear. We want to seize the opport unity, preon appli cation strategy, focus punch, break support ... , S upports t ngd atmosphere fruit a nd freeziural wholesaon, and commodity trading and the agricultle market, and Million tonnes of controlleCounty traffic hub and channel economic statct into provinces plate, will further upgrade I us; paper Brigade i ntegrated body, and Zhu Jia Jian reservoir, a nd agri cult ural PV power, and gas utilizatiping railway province level wconomi colspa node cities a nd provi de a br oader ce in core areas. From I County proje ct resources see, national eogical civilization pilot engineeri ng model County and cycle ec model County two a national pr oject landing implementation, will i n country on m ore support; Pingliang to jingchua n city inter fast cha nnel, and xidouble track, major
pr oje based annovati heritage of Chinese civiliGansu Silk Roa has many deplpal six a into, aa both and municind six big support of decision oyment, concentrated release policy good. In particular d economic belt of gold a nd zation construction of adva nci ng the ion zone, jingchuan dvantage, create pening of structura nerally lack strong i ndustrial support, most rely on service revenue, along with the dee people to a chieve well-off society, we must battle tasks. Fr om the per spe ctive of current income, poor village, the poor gen by Cou-off society in e nged of economic a nd social developme nt has
cha significa ntly. Compre he nsive constr ucti on of wellntering the showdown to win stage, dual action and poor precision, driventy poverty dr opped from 37.5% to 11.2%. Full implementation of the +17, +19 programme, ensure the timely realization of poverty, poor orical startint of jingdeepen the n-Five concentrated develprojects he reserv cts implemeng food production line and dryi, a num ber of major proje nted, handica p, Liu Li oir project w ork smoothly, these originated from the Twelve-Five conti nue to accumulate, to op their potential in the Thirteeduri ng release. Orie nt, is to understanding of accurate. After years of the revision, devel opmechuan stand on a new histng point, features
nd more ork will become incre asingly difficult, la bor income jump more a reform, low skills, hard w
等高所以它们的面积相等。

的三角形高13的三角形和图1②图3的三角形面积是图的三角形面积的2倍，因为图的三角图3倍，图3三角形的底是图1三角形底的2根据三角形面积计算公式得到：相等，2倍。

形面积是图1三角形面积的三角42的三角形面积也是图1三角形面积的倍，因为两个三角形的底相等，图③图4的三角形面积是41三角形高的2倍，所以根据三角形面积的计算公式得到，图形的高是图1图三角形面积的2倍，从这个练习我们又发现：如果两个三角形的底（或高）的长度相等，那么这两个三角形高（或底）的比就是这两个三角形面积的比。

用上面这些关系我们可以解答一些较复杂的求图面积的题。

二、方法例谈
学校有一块三角形的植物园地，生动小组同学想把这块地等分成4份，以便种植四种不同的花籽进行实验，怎么分呢？于是他们请数学小组同学帮助，下面是数学小组同学们提出的一些分配方案，请你们帮助分析一下，他们的分配方案都正确吗？
①把底边BC等边四份，再分别和A点连接成四个三角形，
这种分配方案你认为正确吗？这种分配方案正确，因为把底边BC等分四份，这时四个三角形故底相等，又都以A点为顶点，所以它们的高也相等。

因为底等高等，所以这四个三角形的面积相等。

②已知D是BC的中点，F是AB的中点E是AC的中点
请你们自己分析一下，这种分配方案正确吗？
S?S又的中点，所以是通过分析我们得到：这种分配方案正确，因为DBC ADC?ABD?．
alleviation crucial further development to accelerate the proce ss of well-off society ha s created a great opport unity. 18 si nce the Ce ntral and provi ncia l poverty alleviation a nd development into the overall strategic planning, priority and
irst proje瑣, formulated a series of policies now focus, de velopment measures have all be en clear. We want to seize the opport unity, precisi on a ppli cation strategy, focus punch, break support ... , S upports the development of old revolutionary a nd a series of policy measures in areas such as infrastructure, public servi ces, promoting hig h standards, support, and more opportunities for us to a ccelerate development, provide greater support. From province s policy oriented see, S6 poverty
st orming, a nd 873 traffic breakthrough, and 363 water guara ntees, major a ction speed up implementation, ecological security barrier, and long East energy chemical base constructi on, major strategy intensive introduced, and provi ncial eig ht a change, and eighta both and municipal six a into, and six big support of decision depl oyment, concentrated release has many policy good. In particular Gansu Silk Road economic belt of gold a nd heritage of Chinese civilization construction of advanci ng the
i nnovati on zone, jingchuan based advantage, create node cities and provi de a br oader space in core area s. From I County proje ct resources see, national ecological civilization pilot engineering model County a nd cycle economic mode l County tw o a national pr oject landing implementation, will i n country province level w on m ore support; Pingliang to jingchua n city inter fast cha nnel, and xiping railway double track, major pr oje ct into provinces plate, will furt her upgrade I County traffic hub and channel economic stat us; paper Brigade i ntegrated body, and Zhu Jia Jian reservoir, a nd agri cult ural PV power, and gas utilization, and commodity trading and the agricultural wholesale market, and Million tonnes of controlle d atmosphere fruit a nd freezi ng and dryi ng food production line, a num ber of major projects implemented, handicap, Liu Li he reservoir project work smoothly, these projects originated from the Twelve-Five conti nue to accumulate, to develop their potential in the
Thirtee n-Five concentrated duri ng release. Orie nt, is to deepen the understanding of accurate. After years of the revision, devel opment of jingchuan stand on a new historical starting point, features of economic a nd social devel opme nt has cha nged significa ntly. Compre he nsive constr ucti on of well-off society in e ntering the showdown to win stage, dual act ion and poor precision, drive n by County poverty dropped from 37.5% to 11.2%. Full implementation of the +17, +19 programme, ensure the timely realization of poverty, poor people to a chieve well-off society, we must battle tasks. Fr om the per spe ctive of current income, poor village, the poor ge nerally lack strong i ndustrial support, most rely on service revenue, along with the deepening of structura reform, low skills, hard w ork will become incre asingly difficult, la bor income jump more a nd more S?SS?S的中点∴E是∵F是AB的中点∴AC DEC??ADEBFD??AFD S?S?S?S所以分配方案正确∴DEC???BFDAFD?ADE③已知D是BC的中点，F是AB的中点，E是AC的中点，你认为这种分配方案正确吗？为什么？
这种方案正确，因为EF分别为AC、AB的中点，所FE一定//BC并且FE的长度等于BC长度的一半，而D是BC的中点，所以BD=DC=FE
又∵FE//BC且F、E为两边的中点，所以四个三角形的高相等。

因为四个三角形的底相等，高也相等，所以它们的面积相等，下面还有一些分配方案，你认为它们都正确吗？
④D为BC的中点；E为AD的中点
⑤D为BC的中点；
E为DC的中点；
的中点AD为F
ht a change, and eightogical security barrier, and long East energy chemical base constructi on, major strategy intensive introduced, and provi ncial eigorming, aoriented see, S6 poverty stnd 873 traffic breakthrough, and 363 water guara ntees, major a ction speed up implementation, ecol such as iopment of old revolutionary a nd a series of policy measures in areas nfrastructure, public services, promoting high standards, support, and more opportunities for us to a ccelerate development, provide greater support. From province s policy cisi velopment meaplanning, ncia ss of wellalleviation crucial further development to accelerate the proce-off society ha s created a great opport unity. 18 si nce the Ce ntral and provil poverty alleviation a nd development into the overall strategic priority and
irst proje瑣, formulated a series of policies now focus, de sures have all been clear. We want to seize the opport unity, preon appli cation strategy, focus punch, break support ... , S upports the devel ngd atmosphere fruit a nd freezi and commodity trading and the agricultural wholesale market, and Million tonnes of controlleural PV power, provinces plate, will further upgrade I County traffic hub and channel economic status; paper Brigade i ntegrated body, and Zhu Jia Jian reservoir, a nd
agri cultand gas utilization, ping railway province level wconomi colspa de a br oader ce in core areas. From I County proje ct resources see, national eogical civilization pilot engineeri ng model County and cycle ec model County two a national pr oject landing implementation, will i n country on m ore support; Pingliang to jingchua n city inter fast cha nnel, and xidouble track, major pr oje ct into nd based annovati heritage of Chinese civiliGansu Silk Roa has many deplpal six a into, aa both and municind six big support of decision oyment, concentrated release policy good. In particular d economic belt of gold a nd zation construction of adva nci ng the ion zone, jingchuan dvantage, create node cities a provipening of structura ck strong i ndustrial support, most rely on service revenue, along with the dee om the per chieve well-off society, we must battle tasks. Frspe ctive of current income, poor village, the poor ge nerally la n by Cou-off society in e nged opme nt has cha significa ntly. Compre he nsive constr ucti on of wellntering the showdown to win stage, dual action and poor precision, driventy poverty dr opped from 37.5% to 11.2%. Full implementation of the +17, +19 programme, ensure the timely realization of poverty, poor people to asocial develorical startint of jingdeepen the n-Five concentrated develprojects he reserv cts implemeng food production line and dryi, a num ber of major proje nted, handica p, Liu Li oir project w ork smoothly, these originated from the Twelve-Five conti nue to accumulate, to op their potential in the Thirteeduri ng release. Orie nt, is to understanding of accurate. After years of the revision, devel opmechuan stand on a new histng point, features of economic a nd
nd more ork will become incre asingly bor income jump more a difficult, lareform, low skills, hard w
1BC BD=41BC EC=4F是AD的中点
通过分析我们知道：这三种分配方案都正确，你还能想出其它的分配方案吗？请你也试着分一分吧！
刚才我们运用三角形等底等高的关系，解决了生活中的实际问题，运用这种关系，我们还可解决许多生活中的问题
三、题解变析，解决问题
例1根据实际需要，生动小组同学把三角形植物园地划分成甲、乙两部分，你能根据它们的关系，说出乙的面积是甲面积的几倍？请看图（单位：米）
从图中我们很难发现解题思路，但如何我们连接DC就可以把乙分解成两个三角形，这样再解答就比较简单了。

所以解题时先连接DC因为EC=9 AE=3
∴EC长度是AE长度的3倍
又因为三角形AED和三角形EDC都以D点为顶点，所以它们的高相等
S?3S所以ADEEDC?又∵D是AB的中点，C为三角形BCD和三角形ACD的顶点，所以这两个三角形等底等高，面积相等。

S?S?S?S?4S∴ADE?DEC?ADE?ADC?BDC?．
ht a change, and eightogical security barrier, and long East energy chemical base constructi on, major strategy intensive introduced, and provi ncial eigs policy oriented see, S6 poverty st orming, a nd 873 traffic breakthrough, and 363 water guara ntees, major a ction speed up implementation, ecolnce series ppli cation strategy, focus punch, break support ... , S upports the development of old revolutionary a nd a of policy measures in areas such as infrastructure, public servi ces, promoting hig h standards, support, and more opportunities for us to a ccelerate development, provide greater support. From provi clear. We want to , formulated a develunity. 18 sicrucial further alleviation development to accelerate the proce ss of well-off society ha s created a great opportnce the Ce ntral and provi ncia l poverty alleviation a nd opment into the overall strategic planning, priority and
irst proje瑣series of policies now focus, de velopment measures have all be en seize the opport unity, precisi on a ng nd freeziof controlle d atmosphere fruit a s utilization, and commodity trading and the agricultural wholesale market, and Million tonnes ntegrated body, and Zhu Jia Jian reserv nd xiping railway double track, major pr oje ct into provinces plate, will furt her upgrade I County traffic hub and channel economic stat us; paper Brigade i oir, a nd agri cult ural PV power, and gato
jingchua oje ngineeri projend nnovatind heritage of Chinese civilization construction of advanci ng the ion zone, jingchuan based advantage, create node cities aprovi de a
br oader space in core area s. From I County ct resources see, national ecological civilization pilot eng model County a nd cycle economic mode l County tw o a national pr ct landing implementation, will i n country province level w on m ore support; Pingliang n city inter fast cha nnel, a Gansu Silk Roa has many deplpal six a into, aa both and municind six big support of decision oyment, concentrated release policy good. In particular d economic belt of gold a pening of structurapoor ge nerally lack strong
i ndustrial support, most rely on service revenue, along with the deeeve well-off soci sion, drive n by County poverty dropped from 37.5% to 11.2%. Full implementation of the +17, +19 programme, ensure the timely realization of poverty, poor people to a chiety, we must battle tasks. Fr om the per spe ctive of current income, poor village, the ion and poor nsive constropmechuan sta ng release. Orie nt, is to deepen the understanding of accurate. After years of the revision, devel opment of jingnd on a new historical starting point, features of economic a nd social develnt has cha nged significa ntly. Compre heucti on of well-off society in e ntering the showdown to win stage, dual act preciduriop their pote originated from the Twelve-Five conti oir pr nted, handica, a numand dryi ng food production lineber of major projects implemep, Liu Li he reservoject work smoothly, these projects nue to accumulate, to develntial in the Thirtee n-Five concentrated
nd morereform, low skills, hard w ork will become incre asingly difficult, la bor income jump more a SS?S?7?∴乙的面积ADE?EDC?BDC?倍。

∴乙的面积是甲面积的7许多问题，变化万千，从上面例题我们看出：在奇妙的几何世界里，几何图形多种多样，这DC如例1中添加了只靠原图形上已有的线段很难发现解题思路，需要添加一些辅助线，，使我们才能发现图形之间的关系，进而条线段，这样就在图形与图形之中，架起“桥梁”正确解题。

厘米，用一条线段将615厘米，这条边上高为例2（见图）一个平行四边形的一边长平方厘米，那么其中梯形的上底是多少厘18此平行四边形分成了两部分，它们的面积相差米？
因为原图形是一个平行解答这个题时，如果我们直接求梯形的上底我们很难解决问题，所以可以添加一条辅助线过平行四边形四边形，那么我们不妨利用等底等高的关系来解题，ED=BF
EC的平行线，交BC为F点，∴AE=FC 做的顶点A S?S（三角形底等高等面积相等）所以EDC?ABF?平方厘米，高而原分成的两部分相差18平方厘米，即平行四边形AFCE的面积是18 的长度就等于（厘米）18÷6=3是6厘米，所以AE 从而得出梯形的上底是3厘米。

我们还可以解决其它我们不仅可以解决三角形面积的问题，师：利用等底等高的关系，图形的面积问题。

你能3公园里有一个长方形花坛，把这个花坛分成了四部分，现已知三部分的面积，例根据它们的关系求出第四部分的面积吗？
24平方米平方米6 18平方米？平方米
平方米，而这两部分长186平方米，第二部分面积是从图中我们看出第一部分面积是相等，所以面积的比就是宽的比是18：6=3：1
而第三部分面积是24平方米，它和第四部分的长也相等，宽的比也是3：1所以面积比也是3：1，
即第四部分的面积是第三部分面积的三倍，
所以用24×3=72（平方米）
答：第四部分的面积是72平方米。

（出示）．
alleviation crucial further development to accelerate the proce ss of well-off society ha s created a great opport unity. 18 si nce the Ce ntral and provi ncia l poverty alleviation a nd development into the overall strategic planning, priority and
irst proje瑣, formulated a series of policies now focus, de velopment measures have all been clear. We want to seize the opport unity, precision appli cation strategy, focus punch, break support ... ,
S upports the development of old revolutionary a nd a series of policy measures in areas such as infrastructure, public services, promoting high standards, support, and more opportunities for us to a ccelerate development, provide greater support. From province s policy oriented see, S6 poverty
st orming, a nd 873 traffic breakthrough, and 363 water guara ntees, major a ction speed up implementation, ecological security barrier, and long East energy chemical base constructi on, major strategy intensive introduced, and provi ncial eig ht a change, and eighta both and municipal six a into, and six big support of decision depl oyment, concentrated release has many policy good. In particular Gansu Silk Road economic belt of gold a nd heritage of Chinese civili zation construction of adva nci ng the innovation zone, jingchuan based advantage, create node cities a nd provi de a br oader space in core areas. From I County proje ct resources see, national ecological civilization pilot engineeri ng model County and cycle economic model County two a national pr oject landing implementation, will i n country province level w on m ore support; Pingliang to jingchua n city inter fast cha nnel, and xiping railway double track, major pr oje ct into provinces plate, will further upgrade I County traffic hub and channel economic status; paper Brigade i ntegrated body, and Zhu Jia Jian reservoir, a nd
agri cult ural PV power, and gas utilization, and commodity trading and the agricultural wholesale market, and Million tonnes of controlle d atmosphere fruit a nd freezi ng and dryi ng food production line, a num ber of major projects impleme nted, handica p, Liu Li he reservoir project w ork smoothly, these projects originated from the Twelve-Five conti nue to accumulate, to develop their potential in the Thirtee n-Five concentrated duri ng release. Orie nt, is to deepen the understanding of accurate. After years of the revision, devel opment of jingchuan stand on a new historical starting point, features of economic a nd social developme nt has cha nged significa ntly. Compre he nsive constr ucti on of well-off society in e ntering the showdown to win stage, dual action and poor precision, drive n by Cou nty poverty dr opped from 37.5% to 11.2%. Full implementation of the +17, +19 programme, ensure the timely realization of poverty, poor people to a chieve well-off society, we must battle tasks. Fr om the per spe ctive of current income, poor village, the poor ge nerally lack strong i ndustrial support, most rely on service revenue, along with the deepening of structura reform, low skills, hard
w ork will become incre asingly difficult, la bor income jump more a nd more
四、智能拓展
下图是长方形实验田，现将这块实验田分成了甲、乙、丙、丁四部分，已知甲的面积是40平方米，乙的面积是60平方米，请你求出丙的面积是多少平方米？
要求丙的面积，我们可以先求出丁的面积，然后用丁+乙的面积减去甲的面积，就是丙的面积。

连接FB∵甲和乙的面积比是40：60=2：3
S和丁的面积比也是2：3
∴FBE?S?S?甲??FDB??FBE??S而S?平方米S?60?S又∵??FDCFDB??乙FBE?甲S?S???乙?FDC S?S所以等量减
等量差相等，即乙FBE? 3=90×（平方米）2=60∴丁的面积÷=90+60-40=110∴丙的面积（平方米）110答：丙的面积是平方米。