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triangle別????稱Rt△表達(dá)式Rt△ABC提出時間2016.3.10應(yīng)用學(xué)科數(shù)學(xué)適用領(lǐng)域范圍幾何適用領(lǐng)域范圍三角形分類方法按角或邊分類內(nèi)角和度數(shù)180度目錄1?圖示2?判定定理3?特殊性質(zhì)4?判定方法5?基本簡介6?相關(guān)線段7?勾股定理8?應(yīng)用舉例9?斜邊公式10?三角函數(shù)11?解直角三角形圖示直角三角形如圖所示：分為兩種情況，有普通的直角三角形，還有等腰直角三角形（特殊情況)判定定理等腰直角三角形是一種特殊的三角形等腰直角三角形是一種特殊的三角形，具有所有三角形的性質(zhì)：具有穩(wěn)定性、內(nèi)角和為180°。兩直角邊相等，兩銳角為45°，斜邊上中線、角平分線、垂線三線合一，等腰直角三角形斜邊上的高為此三角形外接圓的半徑R。特殊性質(zhì)它除了具有一般三角形的性質(zhì)外，具有一些特殊的性質(zhì)：性質(zhì)1:直角三角形兩直角邊的平方和等于斜邊的平方。如圖，∠BAC=90°，則AB2+AC2=BC2（勾股定理)性質(zhì)2:在直角三角形中，兩個銳角互余。如圖，若∠BAC=90°，則∠B+∠C=90°性質(zhì)3：在直角三角形中，斜邊上的中線等于斜邊的一半（即直角三角形的外心位于斜邊的中點，外接圓半徑R=C/2）。該性質(zhì)稱為直角三角形斜邊中線定理。性質(zhì)4:直角三角形的兩直角邊的乘積等于斜邊與斜邊上高的乘積。性質(zhì)5:如圖，Rt△ABC中，∠BAC=90°，AD是斜邊BC上的高，則有射影定理如下：射影定理圖（1）（AD)2=BD·DC。（2）（AB)2=BD·BC。（3）（AC)2=CD·BC。射影定理，又稱“歐幾里德定理”：在直角三角形中，斜邊上的高是兩條直角邊在斜邊射影的比例中項，每一條直角邊又是這條直角邊在斜邊上的射影和斜邊的比例中項。是數(shù)學(xué)圖形計算的重要定理。性質(zhì)6:在直角三角形中，如果有一個銳角等于30°，那么它所對的直角邊等于斜邊的一半。在直角三角形中，如果有一條直角邊等于斜邊的一半，那么這條直角邊所對的銳角等于30°。證明方法多種，下面采取較簡單的幾何證法。先證明定理的前半部分，Rt△ABC中，∠ACB=90°，∠A=30°，那么BC=AB/2∵∠A=30°∴∠B=60°（直角三角形兩銳角互余）取AB中點D，連接CD，根據(jù)直角三角形斜邊中線定理可知CD=BD∴△BCD是等邊三角形（有一個角是60°的等腰三角形是等邊三角形）∴BC=BD=AB/2再證明定理的后半部分，Rt△ABC中，∠ACB=90°，BC=AB/2，那么∠A=30°取AB中點D，連接CD,那么CD=BD=AB/2（直角三角形斜邊上的中線等于斜邊的一半）又∵BC=AB/2∴BC=CD=BD∴∠B=60°∴∠A=30°性質(zhì)7:如圖，在Rt△ABC中∠BAC=90°，AD是斜邊上的高，則：證明：S△ABC=1/2*AB*AC=1/2*AD*BC兩邊乘以2，再平方得AB2*AC2=AD2*BC2運用勾股定理，再兩邊除以??,最終化簡即得性質(zhì)8:直角三角形被斜邊上的高分成的兩個直角三角形和原三角形相似。判定方法判定1：有一個角為90°的三角形是直角三角形。判定2：若??，則以a、b、c為邊的三角形是以c為斜邊的直角三角形（勾股定理的逆定理）。判定3：若一個三角形30°內(nèi)角所對的邊是某一邊的一半，則這個三角形是以這條長邊為斜邊的直角三角形。判定4：兩個銳角互為余角（兩角相加等于90°）的三角形是直角三角形。判定5：若兩直線相交且它們的斜率之積互為負(fù)倒數(shù)，則兩直線互相垂直。那么這個三角形為直角三角形。判定6：若在一個三角形中一邊上的中線等于其所在邊的一半，那么這個三角形為直角三角形。參考直角三角形斜邊中線定理判定7：一個三角形30°角所對的邊等于某一鄰邊的一半，則這個三角形為直角三角形。判定3和7的證明：已知△ABC中，∠A=30°，∠A，∠C對的邊分別為a，c，且a=??c。求證∠C=90°證法1：正弦定理，在△ABC中，有a:sinA=c:sinC將a與c的關(guān)系及∠A的度數(shù)代入之后化簡得sinC=1又∵0∴∠C=90°證法2反證法，假設(shè)∠ACB≠90°，過B作BD⊥AC于D在Rt△ABD中,∵∠ADB=90°,∠A=30°∴BD=??AB（30°的直角邊等于斜邊的一半）又∵BC=??AB∴BC=BD但BD是B到直線AC的垂線段，根據(jù)垂線段最短可知BD（或從BC=BD得∠BCD=∠BDC=90°，那么△BCD中就有兩個直角，這是不可能的事情）∴假設(shè)不成立，∠ACB=90°證法3利用三角形的外接圓證明作△ABC的外接圓，設(shè)圓心為O，連接OC,OB∵∠BAC=30°，A在圓上∴∠BOC=60°∵OB=OC=半徑r∴△BOC是等邊三角形,BC=OC=r又∵AB=2BC=2r∴AB是直徑∴∠ACB=90°(直徑所對的圓周角是直角)基本簡介等腰直角三角形的邊角之間的關(guān)系 ：（1）三角形三內(nèi)角和等于180°；（2）三角形的一個外角等于和它不相鄰的兩個內(nèi)角之和；（3）三角形的一角大于任何一個和它不相鄰的內(nèi)角；（4）三角形兩邊之和大于第三邊，兩邊之差小于第三邊；（5）在同一個三角形內(nèi)，大邊對大角，大角對大邊.等腰直角三角形中的四條特殊的線段：角平分線，中線，高，中位線.（1）三角形的角平分線的交點叫做三角形的內(nèi)心，它是三角形內(nèi)切圓的圓心，它到各邊的距離相等.（三角形的外接圓圓心，即外心，是三角形三邊的垂直平分線的交點，它到三個頂點的距離相等）.（2）三角形的三條中線的交點叫三角形的重心，它到每個頂點的距離等于它到對邊中點的距離的2倍。（3）三角形的三條高的交點叫做三角形的垂心。（4）三角形的中位線平行于第三邊且等于第三邊的二分之一。（5）三角形的一條內(nèi)角平分線與兩條外角平分線交于一點，該點即為三角形的旁心。注意：①任意三角形的內(nèi)心、重心都在三角形的內(nèi)部 .②鈍角三角形垂心、外心在三角形外部。③直角三角形垂心、外心在三角形的邊上。（直角三角形的垂心為直角頂點，外心為斜邊中點。）④銳角三角形垂心、外心在三角形內(nèi)部。⑤任意三角形的旁心一定在三角形的外部。相關(guān)線段中線：頂點與對邊中點的連線，平分三角形。角平分線：平分三角形一內(nèi)角的線段。高線：三角形中一頂點向?qū)呑鞯拇咕€勾股定理勾股定理如果直角三角形兩直角邊分別為A，B，斜邊為C，那么 A^2+B^2=C^2;； 即直角三角形兩直角邊長的平方和等于斜邊長的平方。如果三角形的三條邊A，B，C滿足A^2+B^2=C^2;，還有變形公式：??，如：一條直角邊是a，另一條直角邊是b，如果a的平方與b的平方和等于斜邊c的平方那么這個三角形是直角三角形。（稱勾股定理的逆定理）由畢達(dá)哥拉斯趙爽在公元前550年提出。應(yīng)用舉例如圖1，是屋架設(shè)計圖的一部分，點D是斜梁AB的中點立柱為BC，DE垂直于橫梁AC，AB=7.4m，∠A=30°，求BC、DE要多長？解：∵DE⊥AC，BC⊥AC，∠A=30°又?斜邊公式（一）已知兩條直角邊的長度 ，可按公式： 計算斜邊。（二）如已知一條直角邊和一個銳角，可用直角三角函數(shù)計算斜邊。直角三角形ABC的六個元素中除直角C外，其余五個元素有如下關(guān)系：∠A+∠B=90°sinA=（∠A的）對邊/斜邊cosA=（∠A的）鄰邊/斜邊tanA=（∠A的）對邊/鄰邊例：角A等于30°，角A的對邊是4米，計算斜邊C是多少？查表sin30°=0.5，斜邊C=4/0.5=8米三角函數(shù)三角函數(shù)值除了查表，也可以用電腦系統(tǒng)自帶的計算器，計算。開始——程序——附件——計算器。這個計算器有兩種模式，點‘查看’有一個下拉菜單，有標(biāo)準(zhǔn)型和科學(xué)型，選擇科學(xué)型，輸入度數(shù)后正弦點sin，余弦點cos，正切點tan，值就直接顯示出來了。這里有一個度和度分秒轉(zhuǎn)換的問題。如 18.69度，其中整數(shù)18就是18°，那么18.69-18=0.69，用0.69×60=41.4這里整數(shù)41就是41分，再41.4-41=0.4，再用0.4×60=24這個24就是秒。18.69度=18度41分24秒也可以用計算器直接轉(zhuǎn)換：輸入度數(shù)18.69——鉤上Hyp——再點dms就顯示出18.4124，這就是18度41分24秒。如要轉(zhuǎn)換回去就輸入18.4124——鉤上Inv——再點dms,就轉(zhuǎn)換了。有一點請注意，顯示度分秒時，小數(shù)點后面是一位數(shù)或三位數(shù)如：15.3； 15.302，應(yīng)讀作15度30分；和15度30分20秒。解直角三角形含義：一般地，直角三角形中，除直角外，共有五個元素，即3條邊和2個銳角，由直角三角形中除直角外的已知元素，求出未知元素的過程，叫做解直角三角形。1.三條邊的關(guān)系：?2.歸納利用解直角三角形的知識解決實際問題的一般過程：將實際問題抽象為數(shù)學(xué)問題（畫出平面圖形，轉(zhuǎn)化為解直角三角形的問題）。根據(jù)條件的特點，適當(dāng)選用銳角三角形函數(shù)等去解直角三角形；得到數(shù)學(xué)問題的答案。得到實際問題的答案。詞條圖冊詞條圖片(8)數(shù)學(xué)名詞A-F?八邊形?八面體?百分比?百分點?百分位數(shù)?半徑?半球?半圓?被乘數(shù)?被除數(shù)?被加數(shù)?被減數(shù)?比?比例?邊?變量?標(biāo)準(zhǔn)差?表面積?并集?補(bǔ)集?不等邊三角形?不等式?不定積分?差?長?常量?乘?乘方?乘數(shù)?除?除數(shù)?垂心?次方?次方根?大于?大于等于?代數(shù)?單調(diào)性?單項式?導(dǎo)數(shù)?等邊三角形?等式方程式?等腰三角形?等腰梯形?等于?底?底面?點?定積分?定理?定義域?對數(shù)?鈍角?鈍角三角形?多邊形?多面體?二次方程?多項式?二次方根平方根?二次方平方?二進(jìn)制?二十面體?反余割?反余切?反余弦?反正割?反正切?反正弦?方差?非正態(tài)分布?分布?分母?分?jǐn)?shù)?分子?負(fù)?復(fù)數(shù)以上名詞按中文名拼音首字母順序排列G-LM-RS-Z"}