2. As shown in the figure, P is a point on the chord AB, CP⊥OP crosses ⊙O at point C, AB=8, AP: Pb = 1: 3, and find the length of PC.
Solution: Shall we draw three lines first? Od? OA? 、OC? Jean Od⊥AB
Get the right triangle OAd,? Right triangle OPd, right triangle OPC
AB = 8,AP:PB= 1:3
∴AP:(AP+PB)= 1:( 1+3)? [ratio theorem]
that is
AP:AB= 1:4
AP=AB/4=8/4=2。
BP=AB-AP=8-2=6,
∵Od⊥AB
∴Ad=Bd (vertical diameter chord splitting theorem)
It can be concluded that d is the midpoint of straight line AB.
AB=8 then AD=AB/2=4.
Then, according to the Pythagorean theorem of the right triangle edge
Right triangle oad: OA 2 = OD 2+4 2
? oa^2=od^2+ 16
Right triangle OPD: OP 2 = OD 2+2 * 2
? od^2=op^2-4
Right triangle OPC: oc 2 = op 2+PC 2
? pc^2=oc^2-op^2
Circle o radius OA=OC:? pc^2=od^2+ 16-op^2
? pc^2=(op^2-4)+ 16-op^2
? pc^2= 12
? pc=√? 12=2√3≈2× 1.732
? pc≈3.46