(1)∵△AOB是等边三角形,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082301292491914679.png)
∴OB=OA=AB=4,∠BOA=∠OAB=∠ABO=60°.
∵PD⊥OB,∴∠PDO=90°,∴∠OPD=30°,∴OD=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012924935338.png)
OP.∵OP=t,∴OD=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012924935338.png)
t,在Rt△OPD中,由勾股定理,得PD=
(2)如图(1)过C作CE⊥OA于E,∴∠PEC=90°,
∵OD=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012924935338.png)
t,∴BD=4-
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012924935338.png)
t.
∵线段BP的中点绕点P按顺时针方向旋转60°得点C,
∴∠BPC=60°.∵∠OPD=30°,
∴∠BPD+∠CPE=90°.∴∠DBP=∠CPE
∴△PCE∽△BPD
∴,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012924997928.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230129250131235.png)
,,
∴CE=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925028518.png)
,PE=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925044496.png)
,OE=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925059555.png)
,∴C(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925059555.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925028518.png)
).
(3)如图(3)当∠PCA=90度时,作CF⊥PA,∴△PCF∽△ACF,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925325709.png)
,∴CF2=PF•AF,
∵PF=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925044496.png)
,AF=4-OF=2-
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925356463.png)
CF=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925028518.png)
,
∴(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925356463.png)
)2=(
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925044496.png)
)(2-
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925356463.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925356463.png)
),
求得t=2,这时P是OA的中点.
如图(2)当∠CAP=90°时,C的横坐标就是4,
∴2+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925356463.png)
=4∴t=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925683411.png)
(4)设C(x,y),
∴x=2+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925356463.png)
,y=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925356463.png)
,∴y=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925730419.png)
x-
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925746528.png)
,
∴C点的运动痕迹是一条线段.当t=0时,C1(2,0),当t=4时,C2(5,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925761344.png)
),∴由两点间的距离公式得:C
1C
2=2
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823012925761344.png)
.