勾股定理逆定理的五种应用
“如果一个三角形的三条边长分别为a、b、c，且有![](https://wkrtcs.bdimg.com/rtcs/image?w=82&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"82","h":"19","dataType":"png","c":"word/media/image1.png","imgOriW":82,"imgOriH":19})
，那么这个三角形是直角三角形。”这就是勾股定理的逆定理。它是初中几何中极其重要的一个定理，有着广泛的应用。下面举例说明。
一. 用于判断三角形的形状
例1. 如图1，![](https://wkrtcs.bdimg.com/rtcs/image?w=45&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"45","h":"17","dataType":"png","c":"word/media/image2.png","imgOriW":45,"imgOriH":17})
中，，，，求证：是直角三角形
![](https://wkrtcs.bdimg.com/rtcs/image?w=110&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"110","h":"126","dataType":"png","c":"word/media/image6.png","imgOriW":110,"imgOriH":126})
证明：由已知得：
![](https://wkrtcs.bdimg.com/rtcs/image?w=84&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"84","h":"22","dataType":"png","c":"word/media/image7.png","imgOriW":83,"imgOriH":22})
，即c是最长边
![](https://wkrtcs.bdimg.com/rtcs/image?w=63&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"63","h":"19","dataType":"png","c":"word/media/image8.png","imgOriW":63,"imgOriH":19})
![](https://wkrtcs.bdimg.com/rtcs/image?w=246&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"246","h":"125","dataType":"png","c":"word/media/image9.png","imgOriW":246,"imgOriH":125})
![](https://wkrtcs.bdimg.com/rtcs/image?w=58&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"58","h":"17","dataType":"png","c":"word/media/image10.png","imgOriW":58,"imgOriH":17})
是直角三角形
二. 用于求角度
例2. 如图2，点P是等边内一点，且，，，求的度数
![](https://wkrtcs.bdimg.com/rtcs/image?w=147&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"147","h":"150","dataType":"png","c":"word/media/image15.png","imgOriW":147,"imgOriH":150})
解：因![](https://wkrtcs.bdimg.com/rtcs/image?w=63&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"63","h":"15","dataType":"png","c":"word/media/image16.png","imgOriW":63,"imgOriH":15})
，以点B为定点，将旋转到达的位置，连结PP”，则
![](https://wkrtcs.bdimg.com/rtcs/image?w=175&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"175","h":"22","dataType":"png","c":"word/media/image21.png","imgOriW":175,"imgOriH":22})
![](https://wkrtcs.bdimg.com/rtcs/image?w=61&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"61","h":"17","dataType":"png","c":"word/media/image22.png","imgOriW":61,"imgOriH":17})
为等边三角形
![](https://wkrtcs.bdimg.com/rtcs/image?w=117&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"117","h":"15","dataType":"png","c":"word/media/image23.png","imgOriW":117,"imgOriH":15})
在![](https://wkrtcs.bdimg.com/rtcs/image?w=48&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"48","h":"17","dataType":"png","c":"word/media/image24.png","imgOriW":48,"imgOriH":17})
中
![](https://wkrtcs.bdimg.com/rtcs/image?w=309&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"309","h":"24","dataType":"png","c":"word/media/image25.png","imgOriW":309,"imgOriH":24})
![](https://wkrtcs.bdimg.com/rtcs/image?w=14&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"14","h":"13","dataType":"png","c":"word/media/image26.png","imgOriW":14,"imgOriH":13})
由勾股定理的逆定理知
![](https://wkrtcs.bdimg.com/rtcs/image?w=297&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"297","h":"19","dataType":"png","c":"word/media/image28.png","imgOriW":297,"imgOriH":19})
三. 用于求边长
例3. 如图3，在中，D是BC边上的点，已知，，，，求DC的长。
![](https://wkrtcs.bdimg.com/rtcs/image?w=134&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"134","h":"126","dataType":"png","c":"word/media/image33.png","imgOriW":134,"imgOriH":126})
解：在![](https://wkrtcs.bdimg.com/rtcs/image?w=46&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"46","h":"17","dataType":"png","c":"word/media/image34.png","imgOriW":46,"imgOriH":17})
中，由可知
又由勾股定理的逆定理知
![](https://wkrtcs.bdimg.com/rtcs/image?w=86&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"86","h":"19","dataType":"png","c":"word/media/image37.png","imgOriW":86,"imgOriH":19})
在![](https://wkrtcs.bdimg.com/rtcs/image?w=62&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"62","h":"17","dataType":"png","c":"word/media/image38.png","imgOriW":62,"imgOriH":17})
中
![](https://wkrtcs.bdimg.com/rtcs/image?w=144&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"144","h":"25","dataType":"png","c":"word/media/image39.png","imgOriW":144,"imgOriH":25})
四. 用于求面积
例4. 如图4，已知![](https://wkrtcs.bdimg.com/rtcs/image?w=57&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"57","h":"17","dataType":"png","c":"word/media/image40.png","imgOriW":57,"imgOriH":17})
，AB＝3，BC＝4，CD＝12，DA＝13。求四边形ABCD的面积。
![](https://wkrtcs.bdimg.com/rtcs/image?w=161&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"161","h":"104","dataType":"png","c":"word/media/image41.png","imgOriW":161,"imgOriH":104})
解：连结AC，在![](https://wkrtcs.bdimg.com/rtcs/image?w=59&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"59","h":"17","dataType":"png","c":"word/media/image42.png","imgOriW":59,"imgOriH":17})
中，由勾股定理得
在![](https://wkrtcs.bdimg.com/rtcs/image?w=46&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"46","h":"17","dataType":"png","c":"word/media/image44.png","imgOriW":46,"imgOriH":17})
中
![](https://wkrtcs.bdimg.com/rtcs/image?w=251&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"251","h":"19","dataType":"png","c":"word/media/image45.png","imgOriW":251,"imgOriH":19})
由勾股定理的逆定理知
![](https://wkrtcs.bdimg.com/rtcs/image?w=86&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"86","h":"19","dataType":"png","c":"word/media/image46.png","imgOriW":86,"imgOriH":19})
![](https://wkrtcs.bdimg.com/rtcs/image?w=183&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"183","h":"24","dataType":"png","c":"word/media/image47.png","imgOriW":183,"imgOriH":24})
![](https://wkrtcs.bdimg.com/rtcs/image?w=153&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"153","h":"61","dataType":"png","c":"word/media/image48.png","imgOriW":153,"imgOriH":61})
五. 用于证明垂直
例5. 如图5，已知正方形ABCD中，![](https://wkrtcs.bdimg.com/rtcs/image?w=65&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"65","h":"15","dataType":"png","c":"word/media/image49.png","imgOriW":65,"imgOriH":15})
，，求证：
![](https://wkrtcs.bdimg.com/rtcs/image?w=149&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"149","h":"168","dataType":"png","c":"word/media/image52.png","imgOriW":149,"imgOriH":168})
证明：连结FC，设AF＝1，则DF＝3，![](https://wkrtcs.bdimg.com/rtcs/image?w=90&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"90","h":"15","dataType":"png","c":"word/media/image53.png","imgOriW":90,"imgOriH":15})
，
在![](https://wkrtcs.bdimg.com/rtcs/image?w=59&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"59","h":"17","dataType":"png","c":"word/media/image55.png","imgOriW":59,"imgOriH":17})
、、中
![](https://wkrtcs.bdimg.com/rtcs/image?w=217&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"217","h":"73","dataType":"png","c":"word/media/image58.png","imgOriW":217,"imgOriH":73})
![](https://wkrtcs.bdimg.com/rtcs/image?w=227&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"227","h":"20","dataType":"png","c":"word/media/image59.png","imgOriW":227,"imgOriH":19})
![](https://wkrtcs.bdimg.com/rtcs/image?w=12&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"12","h":"12","dataType":"png","c":"word/media/image26.png","imgOriW":14,"imgOriH":13})
由勾股定理的逆定理知
即![](https://wkrtcs.bdimg.com/rtcs/image?w=55&md5sum=91c10873a7e15e61e8f2222dc19d49ad&sign=726b55d954&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=70&ipr={"t":"img","w":"55","h":"14","dataType":"png","c":"word/media/image51.png","imgOriW":57,"imgOriH":17})
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