△ABC中,DE∥BC,BC=8,且S△ADE:S△ABC=1:4,那么DE= .
【答案】
分析:由ED與BC平行,根據(jù)兩直線平行得到兩對(duì)同位角相等,進(jìn)而由兩對(duì)角相等得到兩三角形相似,又相似三角形的面積之比等于相似比即對(duì)應(yīng)邊之比的平方,故由已知的面積之比開(kāi)方求出三角形的相似比,即對(duì)應(yīng)邊之比,然后由DE與BC為一對(duì)對(duì)應(yīng)邊,故比值等于相似比,根據(jù)BC的長(zhǎng)即可求出DE的長(zhǎng).
解答:![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161740437736535/SYS201310221617404377365009_DA/images0.png)
解:∵DE∥BC,
∴∠AED=∠ABC,∠ADE=∠ACB,
∴△AED∽△ABC,
又∵
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161740437736535/SYS201310221617404377365009_DA/0.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161740437736535/SYS201310221617404377365009_DA/1.png)
,
∴
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161740437736535/SYS201310221617404377365009_DA/2.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161740437736535/SYS201310221617404377365009_DA/3.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161740437736535/SYS201310221617404377365009_DA/4.png)
=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022161740437736535/SYS201310221617404377365009_DA/5.png)
,
又BC=8,
∴ED=4.
故答案為:4.
點(diǎn)評(píng):此題考查了相似三角形的判定與性質(zhì),理解相似比即為相似三角形的對(duì)應(yīng)邊之比,相似三角形的面積之比等于相似比的平方.掌握相似三角形的面積之比與相似比的關(guān)系是解本題的關(guān)鍵.