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triangle别    称Rt△表达式Rt△ABC提出时间2016.3.10应用学科数学适用领域范围几何适用领域范围三角形分类方法按角或边分类内角和度数180度目录1 图示2 判定定理3 特殊性质4 判定方法5 基本简介6 相关线段7 勾股定理8 应用举例9 斜边公式10 三角函数11 解直角三角形图示直角三角形如图所示：分为两种情况，有普通的直角三角形，还有等腰直角三角形（特殊情况)判定定理等腰直角三角形是一种特殊的三角形等腰直角三角形是一种特殊的三角形，具有所有三角形的性质：具有稳定性、内角和为180°。两直角边相等，两锐角为45°，斜边上中线、角平分线、垂线三线合一，等腰直角三角形斜边上的高为此三角形外接圆的半径R。特殊性质它除了具有一般三角形的性质外，具有一些特殊的性质：性质1:直角三角形两直角边的平方和等于斜边的平方。如图，∠BAC=90°，则AB2+AC2=BC2（勾股定理)性质2:在直角三角形中，两个锐角互余。如图，若∠BAC=90°，则∠B+∠C=90°性质3：在直角三角形中，斜边上的中线等于斜边的一半（即直角三角形的外心位于斜边的中点，外接圆半径R=C/2）。该性质称为直角三角形斜边中线定理。性质4:直角三角形的两直角边的乘积等于斜边与斜边上高的乘积。性质5:如图，Rt△ABC中，∠BAC=90°，AD是斜边BC上的高，则有射影定理如下：射影定理图（1）（AD)2=BD·DC。（2）（AB)2=BD·BC。（3）（AC)2=CD·BC。射影定理，又称“欧几里德定理”：在直角三角形中，斜边上的高是两条直角边在斜边射影的比例中项，每一条直角边又是这条直角边在斜边上的射影和斜边的比例中项。是数学图形计算的重要定理。性质6:在直角三角形中，如果有一个锐角等于30°，那么它所对的直角边等于斜边的一半。在直角三角形中，如果有一条直角边等于斜边的一半，那么这条直角边所对的锐角等于30°。证明方法多种，下面采取较简单的几何证法。先证明定理的前半部分，Rt△ABC中，∠ACB=90°，∠A=30°，那么BC=AB/2∵∠A=30°∴∠B=60°（直角三角形两锐角互余）取AB中点D，连接CD，根据直角三角形斜边中线定理可知CD=BD∴△BCD是等边三角形（有一个角是60°的等腰三角形是等边三角形）∴BC=BD=AB/2再证明定理的后半部分，Rt△ABC中，∠ACB=90°，BC=AB/2，那么∠A=30°取AB中点D，连接CD,那么CD=BD=AB/2（直角三角形斜边上的中线等于斜边的一半）又∵BC=AB/2∴BC=CD=BD∴∠B=60°∴∠A=30°性质7:如图，在Rt△ABC中∠BAC=90°，AD是斜边上的高，则：证明：S△ABC=1/2*AB*AC=1/2*AD*BC两边乘以2，再平方得AB2*AC2=AD2*BC2运用勾股定理，再两边除以  ,最终化简即得性质8:直角三角形被斜边上的高分成的两个直角三角形和原三角形相似。判定方法判定1：有一个角为90°的三角形是直角三角形。判定2：若  ，则以a、b、c为边的三角形是以c为斜边的直角三角形（勾股定理的逆定理）。判定3：若一个三角形30°内角所对的边是某一边的一半，则这个三角形是以这条长边为斜边的直角三角形。判定4：两个锐角互为余角（两角相加等于90°）的三角形是直角三角形。判定5：若两直线相交且它们的斜率之积互为负倒数，则两直线互相垂直。那么这个三角形为直角三角形。判定6：若在一个三角形中一边上的中线等于其所在边的一半，那么这个三角形为直角三角形。参考直角三角形斜边中线定理判定7：一个三角形30°角所对的边等于某一邻边的一半，则这个三角形为直角三角形。判定3和7的证明：已知△ABC中，∠A=30°，∠A，∠C对的边分别为a，c，且a=  c。求证∠C=90°证法1：正弦定理，在△ABC中，有a:sinA=c:sinC将a与c的关系及∠A的度数代入之后化简得sinC=1又∵0∴∠C=90°证法2反证法，假设∠ACB≠90°，过B作BD⊥AC于D在Rt△ABD中,∵∠ADB=90°,∠A=30°∴BD=  AB（30°的直角边等于斜边的一半）又∵BC=  AB∴BC=BD但BD是B到直线AC的垂线段，根据垂线段最短可知BD（或从BC=BD得∠BCD=∠BDC=90°，那么△BCD中就有两个直角，这是不可能的事情）∴假设不成立，∠ACB=90°证法3利用三角形的外接圆证明作△ABC的外接圆，设圆心为O，连接OC,OB∵∠BAC=30°，A在圆上∴∠BOC=60°∵OB=OC=半径r∴△BOC是等边三角形,BC=OC=r又∵AB=2BC=2r∴AB是直径∴∠ACB=90°(直径所对的圆周角是直角)基本简介等腰直角三角形的边角之间的关系 ：（1）三角形三内角和等于180°；（2）三角形的一个外角等于和它不相邻的两个内角之和；（3）三角形的一角大于任何一个和它不相邻的内角；（4）三角形两边之和大于第三边，两边之差小于第三边；（5）在同一个三角形内，大边对大角，大角对大边.等腰直角三角形中的四条特殊的线段：角平分线，中线，高，中位线.（1）三角形的角平分线的交点叫做三角形的内心，它是三角形内切圆的圆心，它到各边的距离相等.（三角形的外接圆圆心，即外心，是三角形三边的垂直平分线的交点，它到三个顶点的距离相等）.（2）三角形的三条中线的交点叫三角形的重心，它到每个顶点的距离等于它到对边中点的距离的2倍。（3）三角形的三条高的交点叫做三角形的垂心。（4）三角形的中位线平行于第三边且等于第三边的二分之一。（5）三角形的一条内角平分线与两条外角平分线交于一点，该点即为三角形的旁心。注意：①任意三角形的内心、重心都在三角形的内部 .②钝角三角形垂心、外心在三角形外部。③直角三角形垂心、外心在三角形的边上。（直角三角形的垂心为直角顶点，外心为斜边中点。）④锐角三角形垂心、外心在三角形内部。⑤任意三角形的旁心一定在三角形的外部。相关线段中线：顶点与对边中点的连线，平分三角形。角平分线：平分三角形一内角的线段。高线：三角形中一顶点向对边作的垂线勾股定理勾股定理如果直角三角形两直角边分别为A，B，斜边为C，那么 A^2+B^2=C^2;； 即直角三角形两直角边长的平方和等于斜边长的平方。如果三角形的三条边A，B，C满足A^2+B^2=C^2;，还有变形公式：  ，如：一条直角边是a，另一条直角边是b，如果a的平方与b的平方和等于斜边c的平方那么这个三角形是直角三角形。（称勾股定理的逆定理）由毕达哥拉斯赵爽在公元前550年提出。应用举例如图1，是屋架设计图的一部分，点D是斜梁AB的中点立柱为BC，DE垂直于横梁AC，AB=7.4m，∠A=30°，求BC、DE要多长？解：∵DE⊥AC，BC⊥AC，∠A=30°又 斜边公式（一）已知两条直角边的长度 ，可按公式： 计算斜边。（二）如已知一条直角边和一个锐角，可用直角三角函数计算斜边。直角三角形ABC的六个元素中除直角C外，其余五个元素有如下关系：∠A+∠B=90°sinA=（∠A的）对边/斜边cosA=（∠A的）邻边/斜边tanA=（∠A的）对边/邻边例：角A等于30°，角A的对边是4米，计算斜边C是多少？查表sin30°=0.5，斜边C=4/0.5=8米三角函数三角函数值除了查表，也可以用电脑系统自带的计算器，计算。开始——程序——附件——计算器。这个计算器有两种模式，点‘查看’有一个下拉菜单，有标准型和科学型，选择科学型，输入度数后正弦点sin，余弦点cos，正切点tan，值就直接显示出来了。这里有一个度和度分秒转换的问题。如 18.69度，其中整数18就是18°，那么18.69-18=0.69，用0.69×60=41.4这里整数41就是41分，再41.4-41=0.4，再用0.4×60=24这个24就是秒。18.69度=18度41分24秒也可以用计算器直接转换：输入度数18.69——钩上Hyp——再点dms就显示出18.4124，这就是18度41分24秒。如要转换回去就输入18.4124——钩上Inv——再点dms,就转换了。有一点请注意，显示度分秒时，小数点后面是一位数或三位数如：15.3； 15.302，应读作15度30分；和15度30分20秒。解直角三角形含义：一般地，直角三角形中，除直角外，共有五个元素，即3条边和2个锐角，由直角三角形中除直角外的已知元素，求出未知元素的过程，叫做解直角三角形。1.三条边的关系： 2.归纳利用解直角三角形的知识解决实际问题的一般过程：将实际问题抽象为数学问题（画出平面图形，转化为解直角三角形的问题）。根据条件的特点，适当选用锐角三角形函数等去解直角三角形；得到数学问题的答案。得到实际问题的答案。词条图册词条图片(8)数学名词A-F 八边形 八面体 百分比 百分点 百分位数 半径 半球 半圆 被乘数 被除数 被加数 被减数 比 比例 边 变量 标准差 表面积 并集 补集 不等边三角形 不等式 不定积分 差 长 常量 乘 乘方 乘数 除 除数 垂心 次方 次方根 大于 大于等于 代数 单调性 单项式 导数 等边三角形 等式方程式 等腰三角形 等腰梯形 等于 底 底面 点 定积分 定理 定义域 对数 钝角 钝角三角形 多边形 多面体 二次方程 多项式 二次方根平方根 二次方平方 二进制 二十面体 反余割 反余切 反余弦 反正割 反正切 反正弦 方差 非正态分布 分布 分母 分数 分子 负 复数以上名词按中文名拼音首字母顺序排列G-LM-RS-Z"}