梯形的兩條對角線互相垂直,其中一條對角線的長是5cm,梯形的高等于4cm,那么梯形的面積是 .
【答案】
分析:首先根據(jù)題意作圖,然后過點D作DF∥AC交BC的延長線于F,即可得DF⊥BD,四邊形ACFD是平行四邊形,然后根據(jù)勾股定理與射影定理,即可求得BF的長,繼而求得梯形的面積.
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161845899770807/SYS201310221618458997708009_DA/images0.png)
解:根據(jù)題意得:AD∥BC,BD⊥AC,DE⊥BC,BD=5cm,DE=4cm,
過點D作DF∥AC交BC的延長線于F,
∴DF⊥BD,四邊形ACFD是平行四邊形,
∴AD=CF,
∴BE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161845899770807/SYS201310221618458997708009_DA/0.png)
=3,
∵BD
2=BE•BF,
∴BF=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161845899770807/SYS201310221618458997708009_DA/1.png)
,
∴梯形的面積是:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161845899770807/SYS201310221618458997708009_DA/2.png)
(AD+BC)•DE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161845899770807/SYS201310221618458997708009_DA/3.png)
(CF+BC)•DE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161845899770807/SYS201310221618458997708009_DA/4.png)
BF•DE=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161845899770807/SYS201310221618458997708009_DA/5.png)
×
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161845899770807/SYS201310221618458997708009_DA/6.png)
×4=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161845899770807/SYS201310221618458997708009_DA/7.png)
.
故答案為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161845899770807/SYS201310221618458997708009_DA/8.png)
.
點評:此題考查了梯形的性質(zhì),勾股定理與射影定理等知識.此題綜合性較強(qiáng),難度較大,解題的關(guān)鍵是注意數(shù)形結(jié)合思想的應(yīng)用,注意輔助線的作法.