题目内容
(2012•利辛县二模)已知三角形的三边长,求三角形面积,有公式:S=![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_ST/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_ST/1.png)
(1)若已知三角形的三边长分别为2、3、4,试运用公式,计算该三角形的面积S;
(2)现在我们不用以上的公式计算,而运用初中学过的数学知识计算,你能做到吗?请试试.如图,△ABC中AB=7,AC=5,BC=8,求△ABC的面积.(提示:作高AD,设CD=x)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_ST/images2.png)
【答案】分析:(1)代入公式就可以得到.
(2)作高AD,设CD=x,在直角△ABD与直角△ACD中,根据勾股定理就可以得到:则52-x2=AD2=72-(8-x)2,从而解得CD的长,根据三角形的面积公式求解.
解答:解:
(1)当a=2,b=3,c=4时![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_DA/0.png)
S=![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_DA/1.png)
=
;
(2)作高AD,设CD=x,则52-x2=AD2=72-(8-x)2
解得![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_DA/3.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_DA/4.png)
∴
.
点评:一般三角形可以通过作高线转化为解直角三角形的问题来解决.
(2)作高AD,设CD=x,在直角△ABD与直角△ACD中,根据勾股定理就可以得到:则52-x2=AD2=72-(8-x)2,从而解得CD的长,根据三角形的面积公式求解.
解答:解:
(1)当a=2,b=3,c=4时
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_DA/0.png)
S=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_DA/1.png)
=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_DA/2.png)
(2)作高AD,设CD=x,则52-x2=AD2=72-(8-x)2
解得
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_DA/3.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_DA/4.png)
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225759361753041/SYS201310212257593617530020_DA/5.png)
点评:一般三角形可以通过作高线转化为解直角三角形的问题来解决.
![](http://thumb2018.1010pic.com/images/loading.gif)
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