在四邊形ABCD中,∠B=∠C=120°,AB=3,BC=4,CD=5,則此四邊形的面積是 .
【答案】
分析:延長BC,CB 分別作AE⊥EF,DF⊥EF,得梯形AEFD,解△ABE得BE,AE,解△CDF得CF,DF,根據(jù)四邊形ABCD的面積為梯形AEFD的面積減去△ABE的面積減去△CDF的面積可以求解.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/images0.png)
解:延長BC,CB,作AE⊥EF,DF⊥EF,
∵∠B=∠C=120°,
∴∠EBA=∠FCD=60°,
∵AE⊥EF,F(xiàn)D⊥EF,
∴BE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/0.png)
AB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/1.png)
,CF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/2.png)
CD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/3.png)
,
AE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/4.png)
AB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/5.png)
,F(xiàn)D=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/6.png)
CD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/7.png)
,
EF=EB+BC+CF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/8.png)
+4=8,
△ABE的面積為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/9.png)
×AE×EB=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/10.png)
,
△CDF的面積為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/11.png)
×CF×FD=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/12.png)
,
梯形AEFD的面積=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/13.png)
(AE+DF)×EF=16
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/14.png)
,
∴四邊形ABCD的面積為16
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/15.png)
-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/16.png)
-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/17.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/18.png)
.
故答案為
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022163109984392517/SYS201310221631099843925001_DA/19.png)
.
點評:本題考查了勾股定理在直角三角形中的運用,考查了三角形、梯形面積的計算,本題中構造梯形AEFD是解題的關鍵.