题目内容
学习《图形的相似》后,我们可以探索两个直角三角形全等的条件所获得的经验,继续探索两个直角三角形相似的条件.![](http://thumb.1010pic.com/pic1/imagenew2/czsx/4/104964.jpg)
(1)“对于两个直角三角形,满足一边一锐角对应相等,或两直角边对应相等,两个直角三角形全等”,类似地,你可以得到“满足_____,或_____,两个直角三角形相似”;
(2)“满足斜边和一条直角边对应相等的两个直角三角形全等”,类似地,你可以得到满足_____两个直角三角形相似”.请结合下列所给图形,写出已知,并完成说理过程.
已知:如图,_____.试说明Rt△ABC∽Rt△A/B/C/.
(1)一个锐角对应相等,两直角对应成比例
(2)略解析:
(1)一个锐角对应相等,两直角对应成比例;
(2)斜边和一条直角边对应成比例.
在Rt△ABC和Rt△A/B/C/中,∠C=∠C/=90°,
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/6/104966.gif)
解法一:设
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/6/104966.gif)
在Rt△ABC和Rt△A/B/C/中,
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/8/104968.gif)
∴
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/9/104969.gif)
∴Rt△ABC∽Rt△A/B/C/.
解法二:如图,假设AB>A/B/,在AB上截取AB//= A/B/,过点B//作B//C//⊥AC,垂足为C//.
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/5/104965.jpg)
∵∠C=∠AC//B//,∴BC∥B//C//,∴Rt△ABC∽Rt△A/B//C//,
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/10/104970.gif)
∵AB//= A/B/,∴
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/11/104971.gif)
又∵
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/6/104966.gif)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/13/104973.gif)
![](http://thumb.1010pic.com/pic1/imagenew2/czsx/14/104974.gif)
∵AB//= A/B/,∠C=∠AC//B//=90°,
∴Rt△AB//C//≌Rt△A/B/C/,∴Rt△ABC∽Rt△A/B/C/.
![](http://thumb2018.1010pic.com/images/loading.gif)
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