题目内容
(2010•巢湖模拟)将一副三角板按如图①所示的位置摆放,使后两块三角板的直角边AC和MD重合,已知AB=AC=16cm,将△MED绕点A(m)逆时针旋转60°后得到图②,两个三角形重叠(阴影)部分的面积大约是多少?(结果精确到0.1cm,![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_ST/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_ST/images1.png)
【答案】分析:过BC、AD的交点F作FG⊥AC于G,由图形可得,∠FCA=45°,∠FAG=60°,则由BG和两角的正切值表示AC的长,再求得FG的长即可,则三角形重叠(阴影)部分的面积即可求出.
解答:
解:设BC、AD相交于F,过F作FG⊥AC于G.
∵∠BCA=45°,
∴CG=FG.
∵在Rt△AFG中,∠CAD=60°.
∴tan∠GAF=![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_DA/0.png)
∴GA=
FG.
设GC=xcm,
∴FG=xcm,AG=
xcm,
由CG+GA=AC得x+
x=16,
∴x=8(3-
).
∴S△AFC=
AC•FG=
×16×8(3
)≈81.3(cm2).
点评:本题考查了三角函数的应用,需学会用三角函数解直角三角形.
解答:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_DA/images0.png)
∵∠BCA=45°,
∴CG=FG.
∵在Rt△AFG中,∠CAD=60°.
∴tan∠GAF=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_DA/0.png)
∴GA=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_DA/1.png)
设GC=xcm,
∴FG=xcm,AG=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_DA/2.png)
由CG+GA=AC得x+
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_DA/3.png)
∴x=8(3-
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_DA/4.png)
∴S△AFC=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_DA/5.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_DA/6.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021225725747672651/SYS201310212257257476726016_DA/7.png)
点评:本题考查了三角函数的应用,需学会用三角函数解直角三角形.
![](http://thumb2018.1010pic.com/images/loading.gif)
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