题目内容
如图1,已知O是锐角∠XAY的边AX上的动点,以点O为圆心、R为半径的圆与射线AY切于点B,交射线OX于点C,连接BC,作CD⊥BC,交AY于点D.(1)求证:△ABC∽△ACD;
(2)若P是AY上一点,AP=4,且sinA=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_ST/0.png)
①如图2,当点D与点P重合时,求R的值;
②当点D与点P不重合时,试求PD的长(用R表示).
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_ST/images1.png)
【答案】分析:(1)根据切线的性质得到∠ABO=90°,易证∠ABC=∠ACD,从而根据两个角对应相等得到两个三角形相似;
(2)根据(1)中的相似三角形得到对应边的比相等,再结合锐角三角函数的概念,把AD用R表示,①根据AD=AP求得R的值;②应分两种情况讨论,点D可能在点P的左侧或右侧.
解答:(1)证明:由已知,CD⊥BC,
∴∠ADC=90°-∠CBD.
又∵⊙O切AY于点B,
∴OB⊥AB.
∴∠OBC=90°-∠CBD.
∴∠ADC=∠OBC.
又在⊙O中,OB=OC=R,
∴∠OBC=∠ACB.
∴∠ACB=∠ADC.
又∠A=∠A,
∴△ABC∽△ACD.
(2)解:由已知,sinA=
,
又OB=OC=R,OB⊥AB,
∴在Rt△AOB中,AO=
=
R,AB=
=
R.
∴AC=
R+R=
R.
由(1)已证,△ABC∽△ACD,
∴
.
∴
.
因此AD=
R.
①当点D与点P重合时,AD=AP=4,
∴
R=4.
∴R=
.
②当点D与点P不重合时,有以下两种可能:
(i)若点D在线段AP上(即0<R<
),PD=AP-AD=4-
R,
(ii)若点D在射线PY上(即R>
),PD=AD-AP=
R-4,
综上,当点D在线段AP上(即0<R<
)时,PD=4-
R,
当点D在射线PY上(即R>
)时,PD=
R-4,
又当点D与点P重合(即R=
)时,PD=0,故在题设条件下,总有PD=|
R-4|(R>0).
点评:此题要能够熟练运用切线的性质定理、相似三角形的性质和判定.
(2)根据(1)中的相似三角形得到对应边的比相等,再结合锐角三角函数的概念,把AD用R表示,①根据AD=AP求得R的值;②应分两种情况讨论,点D可能在点P的左侧或右侧.
解答:(1)证明:由已知,CD⊥BC,
∴∠ADC=90°-∠CBD.
又∵⊙O切AY于点B,
∴OB⊥AB.
∴∠OBC=90°-∠CBD.
∴∠ADC=∠OBC.
又在⊙O中,OB=OC=R,
∴∠OBC=∠ACB.
∴∠ACB=∠ADC.
又∠A=∠A,
∴△ABC∽△ACD.
(2)解:由已知,sinA=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/0.png)
又OB=OC=R,OB⊥AB,
∴在Rt△AOB中,AO=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/2.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/3.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/4.png)
∴AC=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/5.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/6.png)
由(1)已证,△ABC∽△ACD,
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/7.png)
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/8.png)
因此AD=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/9.png)
①当点D与点P重合时,AD=AP=4,
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/10.png)
∴R=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/11.png)
②当点D与点P不重合时,有以下两种可能:
(i)若点D在线段AP上(即0<R<
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/12.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/13.png)
(ii)若点D在射线PY上(即R>
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/14.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/15.png)
综上,当点D在线段AP上(即0<R<
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/16.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/17.png)
当点D在射线PY上(即R>
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/18.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/19.png)
又当点D与点P重合(即R=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/20.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131103000813800912842/SYS201311030008138009128026_DA/21.png)
点评:此题要能够熟练运用切线的性质定理、相似三角形的性质和判定.
![](http://thumb2018.1010pic.com/images/loading.gif)
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